# Midwest Numerical Analysis Days 2012 May 12-13

Plenary Talks

A Novel, Stable Loosely Coupled Scheme for Fluid-structure Interaction in Hemodynamics

Suncica Canic, Mathematics, University of Houston

The focus of this talk is on modeling and simulation of fluid-structure interaction (FSI) problems in hemodynamics. In particular, the focus is on the interaction between blood flow and arterial wall motion. Blood flow in medium-to-large arteries is modeled by the Navier-Stokes equations for an incomprensible viscous fluid, while the motion of arterial walls will be modeled by a viscoelastic/elastic Koiter shell model.

A set of popular numerical schemes for FSI in blood flow includes the partitioned schemes (loosely or strongly coupled). Partitioned schemes typically solve an underlying multi-physics problem by splitting it into sub-problems determined by the different physics. In particular, in fluid-structure interaction problems the fluid dynamics and structure elastodynamics are solved using separate solvers.

In hemodynamics, the coupling between fluid and structure is highly nonlinear due to the fact that fluid and structure denities are roughly the same. It has been recently shown that in this regime, classical loosely coupled partitioned schemes are unconditionally unstable.

In this talk we will show a novel, loosely coupled partitioned scheme which is unconditionally stable for all the parameters in the problem, including those corresponding to the blood flow application. This scheme is called the kinematically coupled β-scheme. The kinematically coupled β-scheme scheme retains all the advantages of partitioned schemes, such as modularity and easy implementation, while retaining the accuracy of monolithic schemes.

We will prove that this scheme is unconditionally stable and that it converges to a weak solution of the underlying problem. Numerical simulations of a benchmark problem in FSI, as well as physiologically relevant cardiovascular problems, will be presented. Excellent agreement with experimental data will be shown.

This is a joint work with Martina Bukac (UH and U of Pittsburgh), Boris Muha (UH), and Roland Glowinski (UH).

Discontinuous Galerkin Finite Element Methods for High Order Nonlinear Partial Differential Equations

Chi-Wang Shu, Division of Applied Mathematics, Brown University

Discontinuous Galerkin (DG) finite element methods were first designed to solve hyperbolic conservation laws utilizing successful high resolution finite difference and finite volume schemes such as approximate Riemann solvers and nonlinear limiters.  More recently the DG methods have been generalized to solve convection dominated convection-diffusion equations (e.g. high Reynolds number Navier-Stokes equations), convection-dispersion (e.g. KdV equations) and other high order nonlinear wave equations or diffusion equations. In this talk we will first give an introduction to the DG method, emphasizing several key ingredients which made the method popular, and then we will move on to introduce a class of DG methods for solving high order PDEs, termed local DG (LDG) methods.  We will highlight the important ingredient of  the design of LDG schemes, namely the adequate choice of numerical fluxes, and emphasize the stability of the fully nonlinear DG approximations.  Numerical examples will be shown to demonstrate the performance of the DG methods.

No Artificial Numerical Viscosity: From the 1/3 Rule to Entropy Stable Approximations of Navier-Stokes Equations

Eitan Tadmor, Center for Scientific Computation and Mathematical Modeling, University of Maryland

Entropy stability plays an important role in the dynamics of nonlinear systems of conservation laws and related convection-diffusion equations.  What about the corresponding numerical framework? We present a general theory of entropy stability for difference approximations of such nonlinear equations. Our approach is based on comparing numerical viscosities to certain entropy conservative schemes. It yields precise characterizations of entropy stability which is enforced in rarefactions while keeping sharp resolution of shocks.

We demonstrate this approach with a host of first- and second-order accurate schemes ranging from scalar examples to Euler and Navier-Stokes equations. In particular, we present a family of energy-stable schemes for the shallow-water equations, with a well-balanced description of moving equilibria states. Numerical experiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in the immediate neighborhoods of shocks and contacts.

Contributed Talks

Numerical Methods for Shallow Water Equations

Jason Albright, Mathematics, University of Utah

A new well-balanced and positivity preserving scheme for the 2-D shallow water system on a triangular grid will be presented. The proposed method will be compared to existing schemes to illustrate its advantages.

Using a Two-dimensional Continuum Mechanical Model to Predict Collective Cell Migration During Wound Healing

Julia Arciero, Mathematics, Indiana University-Purdue University Indianapolis

Collective cell migration is an important process in wound healing. We have developed a two-dimensional continuum mechanical model to simulate the motion of a cell sheet in response to a wound. The effects of the force generated by lamellipodia, the adhesion between cells and the cell matrix, and the elasticity of the cell layer are included in the model. The partial differential equation describing the evolution of the wound edge is solved numerically using a level set method, and several wound shapes are analyzed. The initial geometry of the simulated wound is defined from the coordinates of an experimental wound taken from cell migration movies. The model is calibrated by comparing the predicted density of the layer and wound edge position with experimentally observed cell density and wound edge position. Experimental observations are combined with mathematical descriptions of cell motion to identify effects of wound shape and area on closure time and to propose a method that uses a simple wound measure (such as area) to predict overall wound healing time early in the closure process.

Superconvergence and a Posteriori Error Estimation for the Local Discontinuous Galerkin (LDG) Method Applied to the Second-order Wave Equation

Mahboub Baccouch, Mathematics, University of Nebraska at Omaha

We analyze the superconvergence properties of the LDG method applied to the second-order wave equation in one space dimension. With a suitable projection of the initial conditions for the LDG scheme, we prove optimal L^2 error estimates for the solution and its spatial derivative. We also prove that the LDG solutions are O(h^{p+2}) super close to particular projections of the exact solutions for p^{th}-degree polynomial spaces. We further show that the significant parts of the discretization errors for the LDG solution and its spatial derivative are proportional to (p+1)-degree right and left Radau polynomials, respectively. These results allow us to prove that p^{th}-degree LDG and its derivative are O(h^{p+2}) superconvergent at the roots of (p+1)-degree right and left Radau polynomials, respectively. Numerical experiments demonstrate that the rates of superconvergence are optimal. These results are used to construct asymptotically correct a
posteriori! error estimates. We prove that the a posteriori LDG error estimates for the solution and its derivative converge to the true spatial errors at O(h^{p+3/2}) rates while computational results show higher O(h^{p+3}) and O(h^{p+2}) convergence rates, respectively. We also prove that the global effectivity indices , for both the solution and its derivative, in the L^2-norm converge to unity at O(h) rate. Several numerical simulations are performed to validate the theory.

Genuinely Multidimensional Riemann Solvers

Dinshaw Balsara, Physics, University of Notre Dame

Arriving at genuinely multidimensional Riemann solvers has been a long-standing quest in CFD. Potentially, such Riemann Solvers offer the promise of greater fidelity with the physics, truly multidimensional wave propagation and larger timesteps.

Some of this promise had been realized in the past. However, the resulting multi-dimensional Riemann solvers were unwieldy and not easy to implement. In this talk I present genuinely multidimensional HLL and HLLC Riemann solvers. I show that they follow from certain natural considerations like self-similarity of the wave model and consistency with the conservation law. Closed form expressions that are easy to implement have been presented in the corresponding paper. This makes multidimensional HLL and HLLC Riemann solvers especially easy to implement.

A Rescaling Scheme for Simulating Diffusional Evolution of Multiple Precipitates in Elastic Medium

Amlan Barua, Applied Mathematics, Illinois Institute of Technology

Simulation of the growth of precipitates in a two phase medium is a fundamental problem in alloy production industry. The primary mechanism of evolution of the precipitates is diffusion but presence of an elastic fields modifies the shape of the precipitates due to the additional elastic energy. Boundary integral equation formulation is a natural choice to solve to this kind of problem. In this talk, a rescaling scheme will be implemented in the boundary integral equations, which leads to exponentially fast numerical simulations and significantly reduces the CPU time. Various numerical simulations with multiple precipitates with be shown to demonstrate the strength of the method.

Parallel Adaptive Tree-code for Computations of Micro Structure in Elastic Medium

Abstract by Hualong Feng: Talk will be presented by Amlan Barua, Applied Mathematics, Illinois Institute of Technology

Three contributions are made in this work. First, a parallel adaptive treecode is developed to simulate evolution of micro-structures in 2D. The code is tested first with randomly generated data for accuracy and time complexity, and then in a boundary integral method for the evolution of micro-structure in elastic media. We document the asymptotic time complexity of the tree-code. Second, the tree-code is applied to compute long-time evolution of micro-structure in elastic media, which would be extremely difficult with a direct summation method due to CPU time constraint. The tree-code speeds up computations drastically while fulfilling the stringent precision requirement prescribed by the spectrally accurate method.  Simulations are presented which show interesting morphology. Third, we propose a new way of parallelizing the treecode. It is shown that the parallel algorithm scales linearly up to a moderate number of processors. The idea of parallelization can be readily used for other treecodes, 2D or 3D, and for uniform or non-uniform point distribution.

Multi-frequency Iterative Method for the Shape Reconstruction of an Acoustically Sound-soft Obstacle Using Multiple Scatterers

Carlos Borges, Mathematical Sciences, Worcester Polytechnic Institute

We consider the problem of determining the shape of an unknown two dimensional acoustically sound-soft obstacle from the measured far field pattern from time-harmonic waves with one fixed incidence and varying frequencies. Starting with a simple initial guess we apply an iterative scheme to obtain an approximate reconstruction of the object at the lowest frequency. This reconstruction is used as an initial guess for the next highest frequency, and this process is repeated until we achieve the result for the highest frequency. At high frequencies, the solution obtained will be sharp in the illuminated part of the object; however, in the shadow part, the reconstruction is not satisfactory. To remedy this setback, we introduce objects with known shape to improve our results. We present numerical results of our method showing the benefits of our approach.

Advantages of Locally Conservative Schemes in Coupled Flow and Transport Models

Steven Brus, Civil Engineering and Geological Sciences, University of Notre Dame

Modeling of the process of sediment transport in shallow water flow has important applications in the design of offshore structures and the prediction of hurricane storm surge. These models require the coupling of equations describing the physics of the flow and transport of sediment. Investigation of the numerical methods used to solve these systems has shown that the use of schemes which are locally conservative has important implications for avoiding artificially induced oscillations in the solution caused by mass conservation errors in the flow. The importance of local mass conservation has been demonstrated by coupling Exner's model for sediment transport with the one dimensional shallow water equations. This coupling was implemented in two different ways. The first uses the commonly used continuous Galerkin (CG) finite element method to solve the flow equations. This method does not conserve mass locally causing errors to occur in the transport component of the solution. The second implementation solves the flow equations with the locally conservative discontinuous Galerkin (DG) finite element method. In this case the elemental mass conservation in the flow results in oscillation free solutions to the transport equation. The development of the two numerical models will be discussed along with the results of a test case to demonstrate the types of errors that can occur when using schemes that are not locally conservative to solve the coupled system.

Modified Shepard's Algorithm for Scattered and Track Data Interpolation

Roberto Cavoretto, Applied Mathematics, Illinois Institute of Technology

Scattered and track data interpolation problems arise in many areas of the applied sciences such as cartography, geophysics and meteorology. Modified Shepard's method for constructing a global interpolant by blending local interpolants using locally supported weight functions usually gives good approximations. Specifically, in this contribution we present a new modified Shepard's algorithm for interpolating large sets of scattered and track data points. This technique is characterized by a suitable partition of the interpolation domain and the use of an efficient nearest neighbor searching procedure in order to find a convenient number of data points in each local neighborhood. Numerical experiments show efficiency and accuracy of the proposed algorithm.

Determining Critical Parameters of Sine-Gordon and Nonlinear Schrödinger Equations with a Point-Like Potential Using Generalized Polynomial Chaos Methods

Debananda Chakraborty, Mathematics, State University of New York at Buffalo

We consider the sine-Gordon and nonlinear Schrödinger equations with a point-like singular source term. The soliton interaction with such a singular potential yields a critical solution behavior. That is, for the given value of the potential strength or the soliton amplitude, there exists a critical velocity of the initial soliton solution, around which the solution is either trapped by or transmitted through the potential. In this talk, we propose an efficient method for finding such a critical velocity by using the generalized polynomial chaos (gPC) method. For the proposed method, we assume that the soliton velocity is a random variable and expand the solution in the random space using the orthogonal polynomials. We consider the Legendre and Hermite chaos with both the Galerkin and collocation formulations. The proposed method finds the critical velocity accurately with spectral convergence. Thus the computational complexity is much reduced. The very core of the proposed method lies in using the mean solution instead of reconstructing the solution. The mean solution converges exponentially while the gPC reconstruction may fail to converge to the right solution due to the Gibbs phenomenon in the random space. Numerical results confirm the accuracy and spectral convergence of the method.

The Effect of Antibody Attachment on a Diffusing Population of Virus

Alex Chen, Mathematical Biology, Statistical and Applied Mathematical Sciences Institute

We study the diffusion of a virus population through a mucus layer and the attachment of antibodies to the surface of individual virions.  Many studies on viral infectivity assume a well-mixed regime of viruses and antibodies, while introducing the virus-antibody mixture directly into a population of cells.  As a result, they tend to overestimate the quantity of antibodies present in the system and ignore the role of antibodies in arresting the diffusion of virus in the mucus layer.

Our study focuses on the interaction of virus and antibodies inside the mucus layer with more physically realistic parameters.  In particular, we will study the distribution of the antibody copy number, the number of attached antibodies to each virion.  We introduce several models, those based purely on stochastic path simulation, those based on a continuum PDE model, and hybrid models that incorporate both path simulation and PDE.  We examine the relative advantages of each model in terms of approximating the true nature of the system and in computation speed.  Several semi-analytical estimates for scaling behavior with respect to various physical parameters in the system are also derived.

Efficient Spectral Methods for Coupled Elliptic Systems with Applications in Phase-field Modeling

Feng Chen, Mathematics, Purdue University, West Lafayette

Many mathematical models for scientific and engineering applications involve linear or nonlinear coupled systems of second-order equations. While spectral methods for single elliptic equations have been well established in many situations, there are few for systems of coupled elliptic equations. How to solve these systems efficiently and accurately present a great challenge, as they usually lead to very ill-conditioned systems due to the high-order derivatives and/or the strong coupling among equations. Another challenge is how to design an algorithm with a complexity that depends linearly on the number of equations. I will talk about how we solve above issues with our new developed spectral-Galerkin methods. Moreover, we designed and implemented a new spectral collocation method for coupled systems that can be parallelized on graphic processing units (GPUs). I will show numerical experiments in isotropic and anisotropic Cahn-Hilliard equations, gradient flow equations from functionalized polymers, and phase-field-crystal equations.

Fast Sweeping Methods for Steady State Problems for Hyperbolic Conservation Laws

Weitao Chen, Mathematics, Ohio State University

Fast sweeping methods are efficient iterative numerical schemes originally designed for solving stationary Hamilton-Jacobi equations. Their efficiency relies on Gauss-Seidel type nonlinear iterations, and a finite number of sweeping directions. In the talk, I will introduce how to generalize the fast sweeping methods to hyperbolic conservation laws with source terms. The algorithm is obtained through finite difference discretization, with the numerical fluxes evaluated in WENO (Weighted Essentially Non-oscillatory) fashion, coupled with Gauss-Seidel iterations. Numerical examples in both scalar and system test problems in one and two dimensions will be shown to demonstrate the efficiency, high order accuracy and the capability of resolving shocks of the proposed methods.

Triangle-based Nodal Discontinuous Galerkin Method for the Shallow Water Equations

Vani Cheruvu, Mathematics and Statistics, University of Toledo

We present a triangle-based nodal discontinuous Galerkin model for the shallow-water equations in two dimensions. The model uses Lagrange polynomials on the triangle and Warburton's interpolation points. We perform a computational study of the convergence rate of the discontinuous Galerkin method in both one and two dimensions for advection and shallow-water equations. The two dimensional models are tested for problems with known analytical solution and the shallow-water model is applied to the generation of Kelvin and gravity waves. The work I mentioned in the abstract is my postdoctoral work with Prof. Max Gunzburger at Florida State University.

Accelerating Spectral Deferred Correction method in ODE

Heejun Choi, Mathematics, Purdue University

Spectral deferred correction (SDC) methods for solving ordinary differential equation (ODEs) were introduced by Dutt, Greengard and Rokhlin (2000). Convergence of SDC is slow for the stiff problem and krylov deferred correction (KDC) was developed by Huang, Jia, Minion (2006) to accelerate SDC. KDC solves defect equation using GMRES with low order (forward or backward Euler) preconditioner. KDC accelerates the convergence but convergence gets slower when underlying ode is stiff. In this paper, we introduce a high order preconditioner which can deal with stiff equation efficiently and it is a direct method for the case of constant coefficient problem. Under some conditions on the equation, the eigenvalues of linearized system tends to cluster to a point if we increase order of method which shows effectiveness of the method.

Computational Modeling Assistant: From Knowledge to Model Specification to GPU Code

Scott Christley, Computer Science, University of Chicago

The translational challenge in biomedical research lies in the effective and efficient transfer of mechanistic knowledge from one biological context to another. Implicit in this process is the establishment of causality from correlation in the form of mechanistic hypotheses. Effectively addressing the translational challenge requires the use of automated methods, including the ability to computationally capture the dynamic aspect of putative hypotheses such that they can be evaluated in a high throughput fashion. Ontologies provide structure and organization to biomedical knowledge; converting these representations into executable models/simulations is the next necessary step. Researchers need the ability to map their conceptual models into a model specification that can be transformed into an executable simulation program. We suggest this mapping process, which approximates certain steps in the development of a computational model, can be expressed as a set of logical rules, and a semi-intelligent computational agent, the Computational Modeling Assistant (CMA), can perform reasoning to develop a plan to achieve the construction of an executable model. We present a prototype implementation for a model construction reasoning process between biomedical and simulation ontologies that is performed by the CMA to produce the specification of an executable model that can be used for dynamic knowledge representation.

Experimental Design for Dynamics Identification of Biological Systems

Vu Dinh, Mathematical Biology, Purdue University

Many problems in systems biology are governed by a dynamical system of differential equations with unknown parameters, where the quantity of interest is the dynamics (time course) of some state variable of the system. The traditional approach to study such systems is designing experiments to estimate values of the parameters from observation. Generally, this problem of parameter identification is an ill-posed inverse problem, and in many case (when the system is lack of identifiability), is unsolvable even if we have an infinite amount of data and even when we restrict the parameter space to a small neighborhood around a nominal parameter values. Another practical issue with parameter identification is computational expense: this problem is a multi-dimensional global optimization problem. In systems biology, the number of system parameters is usually very high, the cost for a single model evaluation is very expensive, the objective function of the optimization problem may contain in finitely many local extrema. Those features make the problem extremely hard and computationally expensive. To overcome these difficulties, in this talk, we propose a novel approach to the problem. Instead of trying to identify the parameters, we derive a method to directly recover the dynamics of the systems from data by constructing a probability distribution on the whole parameters space, based on how well the system controlled by a set of parameter values fits the data. Using this probabilistic framework, we sequentially design the experiment to maximize the amount of information we get about the system form each experiment. This algorithm helps detect the most important features of the dynamics (that need to be measured), then the expected dynamics estimator (EDE) is employed for a complete dynamics recovery.

High Density Waves of the Bacterium Pseudomonas Aeruginosa in Propagating Swarms Result in Efficient Colonization of Surfaces

Huijing Du, Applied and Computational Mathematics and Statistics, University of Notre Dame

The paper describes a new strategy of efficient colonization and community development where bacteria substantially alter their physical environment. Many bacteria move in groups in a mode described as swarming to colonize surfaces and form biofilms to survive external stresses including exposure to antibiotics. One such bacterium is Pseudomonas aeruginosa, which is an opportunistic pathogen responsible for both acute and persistent infections in susceptible individuals, as exampled by those for burn victims and people with cystic fibrosis. P. aeruginosa often, but not always, forms branched tendril patterns during swarming; this phenomena occurs only when bacteria produce rhamnolipid, which is regulated by population-dependent signaling called quorum sensing. The experimental results of this paper demonstrate that P. aeruginosa cells and rhamnolipid propagate as high density waves that move symmetrically as rings within swarms towards the extending tendrils. Biologically-justified cell-based multi-scale model simulations suggest a mechanism of wave propagation as well as branched tendril formation at the edge of the population that depends upon competition between the changing viscosity of the bacterial liquid suspension and the liquid film boundary expansion caused by Marangoni forces. Therefore, P. aeruginosa efficiently colonizes surfaces by controlling the physical forces responsible for expansion of thin liquid films and by propagating towards the tendril tips. The model predictions of wave speed and swarm expansion rate as well as cell alignment in tendrils were confirmed experimentally. The paper results suggest that P. aeruginosa responds to environmental cues on a very short time scale by actively exploiting local physical phenomena to develop communities and efficiently colonize new surfaces.

Upwind-Difference Potentials Method for Patlak-Keller-Segel Chemotaxis Model

Yekaterina Epshteyn, Mathematics, University of Utah

We develop a novel upwind-difference potentials method for the Patlak-Keller-Segel chemotaxis model that can be used to approximate problems in complex geometries. The chemotaxis model under consideration is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration.

Chemotaxis is an important process in many medical and biological applications, including bacteria/cell aggregation and pattern formation mechanisms, as well as tumor growth. Furthermore modeling of real biomedical problems often has to deal with the complex structure of computational domains. There is consequently a need for accurate, fast, and computationally efficient numerical methods for different chemotaxis models that can handle arbitrary geometries.

The upwind-difference potentials method proposed here handles complex domains with the use of only Cartesian meshes, and can be easily combined with fast Poisson solvers. In the presented numerical tests we demonstrate the robustness of the proposed scheme.

First and Second Order Unconditionally Energy Stable Schemes for the Nonlocal Cahn-Hilliard and Allen-Cahn Equations

Zhen Guan, Mathematics, University of Tennessee Knoxville

In this talk I will present unconditionally energy stable schemes for the nonlocal Cahn-Hilliard and Allen-Cahn equations. I will briefly derive the nonlocal models and discuss the difference between the classical (local) and nonlocal CH and AC equations. Stability and convergence theorems of the schemes will be given. Also some numerical simulation results will be provided to show the convergence rates of the scheme and the simulation of the phase separation phenomena.

Super Convergence of Discontinuous Galerkin Method: Eigen-structure Analysis Based on Fourier Approach

Wei Guo, Mathematics, University of Houston

Various super convergence properties of discontinuous Galerkin (DG) method for hyperbolic conservation laws have been investigated in the past. They include the super convergence in negative norm and for properly post-processed solution, the super convergence at Radau and downwind points, the super convergence of dispersion and dissipation error of physically relevant eigenvalues in Fourier analysis and the super convergence toward a special projection of exact solution among many others. Due to these super convergence properties, the DG method has been known to provide good wave resolution properties, especially for long time integrations.

In this talk, via Fourier approach, we observe that the error of DG solution can be decomposed into two parts: (1) the dispersion and dissipation error of the physically relevant eigenvalue; this part of error will grow linearly in time and is of order $2k+1$ (2) projection error, that is, there exists a special projection of the solution such that the numerical solution is much closer to the special projection of exact solution, than the exact solution itself; the magnitude of this part of error will not grow in time.  Based on this fact, we conclude that the error of DG solution will not grow over a period of time that is on the order of $h^{-k}$, where $h$ is the spatial mesh size and $k$ is the degree of polynomial space.  A collection of numerical experiments is presented to demonstrate the analysis.

Solve Nonlinear PDE Using Numerical Algebraic Geometry

Wenrui Hao, Applied and Computational Mathematics and Statistics, University of Notre Dame

Numerical Algebraic Geometry is being used to explore nonlinear partial differential equation system arising from engineering and physics. This new approach is used to compute multiple solutions of nonlinear PDEs and yields the discretized polynomial systems, which involve thousands of variables. This talk will cover the recent progress on nonlinear PDEs based on domain decomposition method. Examples from a tumor growth model will be used to demonstrate the idea.

Subcellular Elements Model for Bacteria Gliding on Graphics Cards

Cameron Harvey, Physics, University of Notre Dame

The collective motion of bacteria is an example of the complex emergent behavior at population scale that arise from decisions that the scale of individual cells. To study the cell-cell interactions that give rise to swarm behavior of gliding bacteria on surfaces, we developed a computational model that captures the biological features of cell flexibility, adhesion, and direction reversals. The model was calibrated and validated with simulations of two-cell collisions in a serial implementation of the model. We then developed a parallel version of the model in CUDA to be run on Graphical Processing Units in order to run simulations of hundreds to thousands of cells.

Parameterized Systems of Equations

Jonathan Hauenstein, Mathematics, Texas A&M University

Systems of both differential equations and analytic equations arising in applications often depend upon parameters.  For some systems, the number of real solutions will change as the parameters vary.  This talk will explore using homotopy techniques to vary the parameters in order to increase (or decrease) the number of real solutions as well as finding parameter values where bifurcations occur.

A Python Finite Element Program Package for Solving Nonlocal Dielectric PDE Models

Yi Jiang, Mathematical Science, University of Wisconsin-Milwaukee

In this talk, I will report a Python program package we developed recently for solving some nonlocal electrostatic continuum models. The package was written based on the Dolfin finite element library and PETSc linear solver library that come from the FEniCS project. I will describe the new splitting algorithm we developed for solving these nonlocal models, and then show some numerical results that verify our new splitting algorithms and display some significant and interesting differences between nonlocal and traditional local dielectric models. This is an ongoing work under the guidance of Professor Dexuan Xie. This work was partially supported by National Science Foundation grant DMS-0921004.

Compressed Sensing with Partially Known Support

Fritz Keinert, Mathematics, Iowa State University

Many signals (such as music, images or videos) are compressible, which means that they can be represented accurately by very few nonzero coefficients in a suitable basis. Examples include JPG compression for images or MPEG for music. Usually one measures the full signal first and then finds the compressed form. The goal of compressed sensing is to determine the signal using a number of measurements comparable to the compressed size of the signal, rather than the full size. This is useful in situations where measurements are expensive or harmful.

In this talk I will give an overview of the mathematical problem, existence/uniqueness results, some numerical approaches that have been tried, and a description of the problem I am currently studying.

Compressed sensing with partially known support assumes that one has an estimate of the location of the nonzero coefficients. For example, in a video sequence one can assume that the image does not change very much from one frame to the next. The goal is to find more efficient algorithms exploiting this a priori knowledge, as well as existence/uniqueness results for this setting.

PDE vs ODE Dynamics

King-Yeung Lam, Mathematics, Ohio State University

Dynamics, or behaviour of solution of nonlinear reaction-diffusion system is deeply related to that of its kinetic problem. In this lecture, we will use the classical Lotka-Volterra competition model as an example to illustrate some connections between the two. Also, the asymptotic behavior of the principal eigenvalue of linear cooperative elliptic systems, as the diffusion rates approach zero, will be motivated and studied.

An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDE with Random Coefficients

Guang Lin, Pacific Northwest National Laboratory

Generalized polynomial chaos (gPC) methods have been successfully applied to various stochastic problems in many physical and engineering fields. However, realistic representation of stochastic inputs associated with various sources of uncertainty often leads to high dimensional representations that are computationally prohibitive for classic gPC methods. Additionally in the classic gPC methods, the gPC bases are determined based on the probabilistic distribution of stochastic inputs. However, the stochastic outputs may not share the same probabilistic distribution as the stochastic inputs. Hence, the gPC bases may not be the optimal bases for such systems, which cause the slow convergence of gPC methods for such stochastic problems. Here we present a general framework that integrates the adaptive ANOVA decomposition technique and the data-driven stochastic method to alleviate both of the two limitations. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique for high-dimensional stochastic problems. Three different ANOVA adaptive criteria are discussed.

To improve the slow convergence of gPC methods, we use the data-driven stochastic method (DDSM), which was developed by Cheng-Hou-Yan in [5]. This method has an offline computation and an online computation. In the offline computation, optimal gPC bases are obtained by Karhunen-Loeve (K-L) expansion of the covariance matrix of stochastic outputs obtained by ANOVA-based sparse-grid PCM. In the online computation, a Galerkin-projection based gPC method with the optimal bases developed in the offline computation is employed, which greatly speeds up the convergence. Numerical examples are presented for one-, two-dimensional elliptic PDE with random coefficients, and a two-dimensional Helmholtz equation in random media (Horn problem) to show the accuracy and efficiency of the developed adaptive ANOVA-based DDSM method.

FDTD Techniques on Overlapping Cells and Moving Meshes

Jinjie Liu, Mathematical Sciences, Delaware State University

A stable anisotropic Finite-Difference Time-Domain (FDTD) algorithm based on overlapping cells is developed for solving Maxwells equations of electrodynamics in anisotropic media. For complex media that include perfectly electric conductor (PEC) and anisotropic dielectric materials, such as the metamaterial cloaking devices, the conventional anisotropic FDTD method suffers from instability due to the interpolation in the finite-difference operator in the constitutive equations. The proposed method is stable as it relies on the overlapping cells to provide the collocated field values without any interpolation.

To simulate long-distance optical pulse propagation, we have developed a moving frame FDTD method with Perfectly Matched Layer (PML) boundary conditions. Good agreement between the moving and stationary FDTD methods is obtained for the pulse propagation in dielectric, dispersive, and nonlinear media.

Some Simple Techniques for the Level-set Interface Capturing with the BFECC Method

Yingjie Liu, Mathematics, Georgia Institute of Technology

Using the BFECC method to compute the level set equation describing a free surface provides a simple implementation of the level set method, because BFECC essentially calls a first order subroutine (such as the unconditionally stable CIR scheme) 3 times. It also has less smearing near sharp corners compared to many other 2nd order methods. In this talk I am going to discuss some techniques for BFECC to deal with non-smooth convection velocity and a simple redistancing procedure.

A Robust Reconstruction for Unstructured WENO Schemes

Yuan Liu, Applied and Computational Mathematics and Statistics, University of Notre Dame

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. In this paper, we combine two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries. Numerical examples including both scalar and system cases are given to demonstrate stability and accuracy of the scheme.

Using Isogeometric Analysis with Non-Matching Discretizations to Simulate Operation of a Pediatric Ventricular Assist Device

Chris Long, Mathematics and Engineering, University of California, San Diego

Congenital heart defects are among the most frequent types of birth defects, and can lead to heart failure in severe cases. Many children awaiting heart transplants will require mechanical circulatory support from a Pediatric Ventricular Assist Device (PVAD) to prolong survival until a donor organ is found. These devices are usually a scaled version of an adult model, and carry high risks of stroke or embolism. These devices can also be useful as a bridge-to-recovery, but risks of thrombosis are an impediment to long-term use.

We have successfully developed a three-dimensional, time-varying finite element framework with Fluid-Structure Interaction to simulate cardiovascular flow. In the present work, we extend this framework to PVAD operation, and implement Isogeometric Analysis on a solid membrane to capture highly non-linear buckling effects. The fluid and solid domains do not have matching discretizations at the shared boundary, and a method for passing boundary conditions across domains is presented. The solid and fluid domains are solved simultaneously with a Matrix Free method used when a traditional stiffness matrix cannot be assembled. Methods and preliminary results are presented.

Transformed Field Expansion for 3D Maxwell Equations

Lina Ma, Mathematics, Purdue University

Inspired by a stable, high-order method for bounded-obstacle acoustic scattering, we develop an efficient and accurate spectral method solving three dimensional Maxwell equations for spherical obstacle in unbounded domain. This high order method is based on transformed field expansion, coupled with spectral-Galerkin solver. To study the Maxwell equations in spherical geometries, we will review some nice formulation of vector spherical harmonics (VSH), which could reduce the 3D problem into 1D proto-typical equations together with a Hemholtz equation in radius direction.

Stable Kernel-based Methods for Approximation

Michael McCourt, Mathematics and Computer Science Division, Argonne National Laboratory

Meshfree kernel-based approximation methods are popular in scattered data interpolation, boundary value problems, statistics, machine learning, and other applications. Significant theory exists regarding the optimality of various kernel methods using radial basis functions (RBF), and because they work on unstructured grids, these methods easily generalize to higher dimensions. Several barriers to their widespread acceptance exist, including computational cost, and numerical instabilities for accurate RBF choices. This work will discuss an eigenfunction expansion for positive definite kernels based on Hilbert-Schmidt theory, which, combined with the RBF-QR work of Fornberg and others, will allow for stable computions with Gaussians in arbitrary dimensions in the traditionally ill-conditioned flat limit. Numerical results regarding interpolation will be presented, and applications in boundary value problems and multiphysics coupling will be discussed. Preliminary work on expansions for the various Matern kernels will also be presented.

Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Conservation Laws in One Space Dimension

Xiong Meng, Applied Mathematics, Brown University

In this paper, the analysis of the superconvergence property of the discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves $(k + \frac32)$-th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree $k$ ($k \ge 1$), under the condition that $|f'(u)|$ possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on $f(u)$ is artificial.

Super Time Stepping for Parabolic and Mixed Systems: The Runge-Kutta-Legendre Method

Chad Meyer, Physics, University of Notre Dame

We present a new method for computing the temporal update for parabolic equations called the Runge-Kutta-Legendre (RKL) method.  It is a well known problem for parabolic systems that the maximum stable time-step drops with the square of the grid size, so that explicit solutions become less and less computationally efficient under grid refinement.  Rather than relying on implicit methods or costly explicit sub-cycling, the RKL method can "accelerate" the parabolic update by computing the operator only n times to achieve a full time update of n^2 times the explicit time-step. The RKL method has been formulated at second order accuracy, and has many additional favorable features such as internal stability and accuracy, and fractional stability, which are not shared by other Super Time Stepping methods.  Additionally, in many important numerical applications, one often encounters systems which mix hyperbolic and parabolic operators, creating a mismatch in the explicit! time-steps between the hyperbolic and parabolic parts of the system. The RKL method can support mixed systems in either an operator split or unsplit manner.

Numerical Studies of the Focusing Davey-Stewartson II and Focusing Klein Gordon Equations

Benson Muite, Mathematics, University of Michigan

We give a brief review of previous results on the Davey-Stewartson and Klein-Gordon equations. We then discuss how simulations using Fourier spectral methods can be used to explore initial data that lead to blow up. Part of this work is joint with Christian Klein and Kristelle Roidot (http://arxiv.org/abs/1112.4043).

A Numerical Study of Traveling Wave Solutions for the Extended Generalized Regularized Long Wave-equation

Sanja Pantic, Mathematics, Statistics, and Computer Science, DePaul University, Chicago

The Regularized Long Wave-equation $u_t+u_x+(u^2)_x-u_{xxt}=0,$ was first studied as a model for small-amplitude long waves   that propagate   on the free surface of a perfect fluid. As an alternative to the Korteweg-de Vries equation, it features a balance between nonlinear $(u^2)_x$ and frequency dispersive term $-u_{xxt}$ that allow existence of traveling waves that are smooth and symmetric about their maximum. Such waves decay rapidly to zero on their outskirts and, because of their tendency to travel alone, are known as   solitary waves. We are interested in solitary-wave solutions of the equation $u_t+ \alpha u_x+\beta_p (u^p)_x+\beta_q (u^q)_x-\gamma u_{xxt}=0,$ which we  call the EGRLW-equation  (Extended  Generalized Regularized Long Wave-equation). Since the solutions are not known analytically the behavior of such solutions, for various powers  $p$ and $q$, where $p<5<q$ as well as different values of the coefficients $\alpha$, $\beta_p$, $\beta_q$ and $\gamma$ is investigated numerically.

Of particular interest is the effect of   nonlinearities and the balance that they strike with dispersion to create the solitary waves. Since the EGRLW-model with general nonlinearities that we are considering does not describe motion of the shallow water as the RLW-equation that it originates from did, its applications go beyond shallow water waves. Stable propagation and interaction of this traveling waves is of interest for many applications in science and engineering. We present numerical simulations of the formation and evolution of solitary waves, the behavior of solitary waves under amplitude perturbation as well as interaction of two solitary waves.

Framework for Stability Analysis of Grid Overlapping Algorithms

Yulia Peet, Engineering Sciences and Applied Mathematics, Northwestern University

With the advance in performance and capabilities of modern computers, interest in integrated large-scale simulations of systems is growing. New class of algorithms is emerging to address the integration of codes, and domain decomposition through overlapping grids is one of the popular developments. Although gaining increased popularity among the engineering community, overlapping grid methods fall behind in numerical analysis of the properties of underlying schemes.

In this talk, we develop a theoretical framework to analyze stability of overlapping grid methods in a situation where interface terms are explicitly extrapolated in time and conventional methods of stability analysis are not directly applicable. We apply our framework to analyze stability of parabolic equations on overlapping grids. We also investigate an influence of inter-grid iterations on stability properties. Numerical experiments are then presented relating theoretical stability bounds to the observed numerical values. Finally, we demonstrate that observed stability trends agree with the behavior of 3D Navier-Stokes solver where the overlapping grid technique was recently implemented.

Use of the String Method to Find Minimal Energy Paths of Droplets on Superhydrophobic Surfaces

Kellen Petersen, Applied Mathematics, Courant Institute, New York University

Interest in superhydrophobic surfaces has increased due to a number of interesting advances in science and engineering. Here we use a diffuse interface model for droplets on topographically and chemically patterned surfaces in the regime where gravity is negligible.  We then apply the constrained string method to examine the transition of droplets between different metastable/stable states.  The string method finds the minimal energy paths (MEPs) which correspond to the most probable transition pathways between the metastable/stable states in the configuration space.  In the case of a hydrophobic surface with posts of variable height and separation, we determine the MEP corresponding to the transition between the Cassie-Baxter and Wenzel states. Additionally, we realize critical droplet morphologies along the MEP associated with saddle points of the free-energy potential and the energy barrier of the free energy. We analyze and compare the MEPs and free-energy barriers for a variety of surface geometries, droplets sizes, and static contact angles ranging from partial wetting to complete wetting.  We also introduce an unbiased double well potential in the diffuse interace model by introducing a chemical potential that is fixed for a given simulation.  We find that the energy barrier shifts toward the Wenzel state along the MEP as the height of the pillars increases in the topographically patterned case while a shorter energy barrier exists and is more centered along the MEP for pillars of shorter height.  More importantly, we demonstrate the string method as a useful tool in the study of droplets on superhydrophobic surfaces by presenting a numerical study that finds MEPs in configuration space, critical droplet morphologies and free-energy barriers which in turn give us a greater understanding of the free-energy landscape.

Numerical Simulation of Self-Assembled Magnetic Colloidal Structures Using GPUs

David Piet, Engineering Sciences and Applied Mathematics, Northwestern University

Self-assembled magnetic colloidal structures that lie at a fluid-air interface can have a wide range of behavior, from localized axisymmetric star-like objects to linear, snake-like ones.  Modeling these stars requires both extensive use of the Navier-Stokes Equations from an analytic standpoint as well as the ability to numerically solve and simulate them alongside Newton's Equations.  Analytically, these equations are approximated using an asymptotic expansion with a small viscosity.  Using those expressions, simulations are run on GPUs to utilize the their parallel capability.  The few hundred cores on these graphics cards allows each particle's behavior to computed in parallel and thus allows for massively scaled simulations to be run over the course hours instead of days or weeks.

Verified and Validated Inviscid and Viscous Calculation of Hydrogen-Air Detonation

Joseph Powers, Aerospace and Mechanical Engineering, University of Notre Dame

The dynamics of one-dimensional hydrogen-air detonations predicted in the inviscid limit as well as with the inclusion of mass, momentum, and energy diffusion were investigated numerically. A series of calculations in which the velocity of a driving piston was varied was performed. Strongly overdriven detonations are stable; as the overdrive is lowered, the longtime behavior of the system becomes more complex. It was found that detonations propagating into a stoichiometric hydrogen-air mixture at 0.421 atm and 293.15 K develop single frequency pulsations at a critical overdrive of f = 1.13. In the inviscid limit using shock-fitting, an oscillation at a frequency of 0.97 MHz was predicted for a f = 1.1 overdriven detonation, which agrees well with the value of 1.04 MHz observed in the equivalent shock-induced combustion experiment around a spherical projectile. The amplitude of these pulsations grows as the overdrive is lowered further. Decreasing the overdrive yet further, a bifurcation process occurs in which modes at a variety of frequencies are excited. The addition of physical mass, momentum, and energy diffusion has a stabilizing effect on overdriven detonations relative to the inviscid limit. In the viscous analog, the structure of these detonations is modulated, and the amplitude of the oscillations can be significantly decreased. Therefore, depending on the application, the use of the reactive Euler equations may be inappropriate, and the reactive Navier-Stokes may be a more appropriate model.

Efficient Numerical Techniques for Adaptive Total Variation Regularization in Image Processing

Surya Prasath, Computer Science, University of Missouri-Columbia

We consider an adaptive version of the well-known total variation (TV) based regularization model in image processing. Adaptive TV based schemes try to avoid the staircasing artifacts associated with TV regularization based energy minimization models. An algorithm based on the modification of the split Bregman technique proposed by Goldstein and Osher[1] can be used for solving the adaptive case. Convergence analysis of such an alternating scheme is proved using the Fenchel duality and a recent result of Svaiter[2] on the weak convergence of Douglas-Rachford splitting method. Moreover, the variational formulation naturally leads to the biased anisotropic diffusion framework. I will provide detailed comparative experimental results using the modified split Bregman, dual minimization and additive operator splitting for the diffusion equation to highlight the efficiency of adaptive TV based schemes for image denoising, segmentation problems.

References:

[1] T. Goldstein, S. Osher, The split Bregman algorithm for L1 regularized
problems, SIAM Journal on Imaging Sciences 2 (2) (2009) 323-343. doi:10.1137/080725891.

[2] B. F. Svaiter, On weak convergence of the Douglas{Rachford method,
SIAM Journal on Control and Optimization 49 (1) (2011) 280-287. doi:10.1137/100788100.

Use of Model Reduction in Multi-fidelity Uncertainty Analysis

Oleg Roderick, Mathematics and Computer Science Division, Argonne

We investigate the issues of uncertainty analysis of complex simulation models. Given that each high-resolution model evaluation is computationally expensive, with the increase in the dimension of the uncertainty space, pure sampling-based methods start to fail due to the curse of dimensionality. We explore the possibilities of hybrid, semi-intrusive uncertainty analysis, based on sampling of the same model at different levels of fidelity, with the majority of evaluations performed at lower fidelity, at much lower computational cost. We use a Gaussian-processes based approach to assess the error in lower-fidelity model evaluations, and to compensate for it when recovering the full model's response to uncertainty. POD-based model reduction is an attractive choice for generation of lower-fidelity data, due to flexibility of the procedure, and the existence of error estimation techniques for the model reduction. Our ongoing work includes applying the approach to complex (Navier-Stokes) flow models used in nuclear engineering.

An ETD-Crank Nicolson Scheme for Reaction-Diffusion Equations

Bruce Wade, Mathematical Sciences, University of Wisconsin-Milwaukee

Exponential Time Differencing (ETD) schemes are highly efficient methods to solve nonlinear reaction-diffusion equations. We will give an idea on the recent development of some effective second order ETD schemes and then proceed to describe applications and examples.

2D Swimming at Low Reynolds Number

Qixuan Wang, School of Mathematics, University of Minnesota

Cell migration is crucial for many biological processes. To date, a lot have been done for cells crawling. We are interested in another mode of migration---self-propelled swimming at low Reynolds number, in which the interaction between the cell and the extracellular matrix is absent. By mathematically generating general shape deformations of planar Stokes flow swimmers, we study those factors that play crucial roles in the swimming process and prescribe what kind of shape deformations may lead to more efficient swimming.

Some Aspects of High-order Discontinuous Galerkin Methods for Shallow Water Equations

Damronsak Wirasaet, Engineering, and Joannes Westerink, Civil Engineering and Geological Sciences, University of Notre Dame

We present an investigation on several aspects of high-order Discontinuous Galerkin (DG) solutions of the two-dimensional nonlinear Shallow Water Equations (SWE). Numerical performance in terms of accuracy and computational cost is comprehensively examined through h- and p- convergence studies on a problem with a manufactured solution. Modal and nodal DG methods on a range of unstructured meshes are considered in this performance study. In addition, we discuss some aspects to be considered in order to gain benefit from using high-order DG methods for simulating more realistic coastal flow scenarios, including flow in open channels.

Randomized Sparse Direct Solvers for Discretized PDEs

Jianlin Xia, Professor, Mathematics, Purdue University

We propose some randomized structured direct solvers for large discretized PDEs, such as Helmholtz equations. New randomization and low-rank techniques are used, together with flexible methods to exploit structures in the factorizations. Our randomized structured techniques provide both higher efficiency and better applicability than some existing structured methods. New fast ways are proposed to conveniently perform various complex operations which are difficult otherwise.

We also study the issues that are closely related, such as matrix-free direct solvers, frequency update, preconditiong, sparse eigenvalue solution, etc. The methods are especially useful for quickly solving 3D problems with multiple frequencies.

A New Analysis of Electrostatic Free Energy Minimization and Poisson-Boltzmann Equation for Protein in Ionic Solvent

Dexuan Xie, Mathematical Sciences, University of Wisconsin-Milwaukee

The nonlinear Poisson-Boltzmann equation (PBE) is one widely-used continuum model for computing the electrostatic potential function of protein in ionic solvent. To analyze its solution existence and uniqueness, one way is to treat it as a linear Poisson dielectric model with optimal ionic charge density functions estimated from the minimization of an electrostatic free energy functional. Several PBE analyses were done in this way. To further improve them, we restudied the Poisson dielectric model and the electrostatic free energy minimization problem recently based on a novel solution decomposition. In particular, the electrostatic free energy functional and the minimization problem are modified equivalently as the new ones without involving any singular function when the integral domain is restricted into the region of ionic solvent. Moreover, the new electrostatic free energy functional is calculated for its first and second Gateaux derivatives, and is then ! expressed in a Taylor expansion form. Consequently, the new minimization problem and PBE can be analyzed directly by the standard techniques of variational methods, including the cases with uniform and nonuniform ionic sizes.  In addition, a new expression of the electrostatic free energy functional is proposed for the calculation of electrostatic free energy. This work sets up a theoretical base and a framework in the development of fast minimization algorithms for solving various PBE-type models numerically. It was partially supported by the National Science Foundation, USA, through grant DMS-0921004.

Angular Distribution of Lyα Resonant Photons Emergent from Optically Thick Medium

Yang Yang, Applied Mathematics, Brown University

We investigate the angular distribution of Lyα photons transferring in or emergent from an optically thick medium. Since the evolutions of specific intensity I in the frequency space and the angular space are coupled from each other, wet develop the WENO numerical solver in order to find the time-dependent solutions of the integro-differential equation of I in the space of frequency and angular simultaneously. We show first that the solutions with the Eddington approximation, which assume I to be linearly dependent on the angular variable µ, yield similar frequency profiles of the photon flux as that without the Eddington approximation. However, the solutions of the µ distribution evolution are significantly different from that given by Eddington approximation. First, the angular distribution of I are found to be substantially dependent on the frequency of photons. For photons with the resonant frequency ν0 , I contains only a linear term of µ. For photons with frequency at the double peaks of the flux, the µ-distribution is highly anisotropic, in which most photons are in the direction of radial forward. Moreover, either at ν0 or at the double peaks, the µ distributions actually are independent of the initial µ distribution of photons of the source.

An Entropy Satisfying Conservative Method for the Fokker-Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model

Hui Yu, Computational and Applied Math, Iowa State University

We propose an entropy satisfying conservative method to solve the Fokker Planck equation of the finitely extensible nonlinear elastic dumbbell model for polymers, subject to homogeneous fluids. Both semidiscrete and fully discrete schemes satisfy all three desired properties (i) mass conservation, (ii) positivity preserving, and (iii) entropy satisfying in the sense that these schemes satisfy discrete entropy inequalities for both the physical entropy and the quadratic entropy. These ensure that the computed solution is a probability density and the schemes are entropy stable and preserve the equilibrium solutions. We also prove convergence of the numerical solution to the equilibrium solution as time becomes large. Zero flux at boundary is naturally incorporated, and boundary behavior is resolved sharply. Both one- and two-dimensional numerical results are provided to demonstrate the good qualities of the scheme and the effects of some canonical homogeneous flows.

An Optimal Convergence Rate Robin-Robin Domain Decomposition Method

Shangyou Zhang, Mathematical Sciences, University of Delaware

In this talk, we shall answer a long-standing question: Is it possible that the convergence rate of the Lions' Robin-Robin nonoverlapping domain decomposition method is independent of the mesh size $h$? The traditional Robin-Robin domain decomposition method converges at a rate of $1-O(h^{1/2})$, even under the optimal parameter. We shall design a two-parameter Robin-Robin domain decomposition method. It is shown that the new DD method is optimal, which means the convergence rate is independent of the mesh size $h$.

Boundary Detection for Singular Perturbation Problems

Weiqun Zhang, Mathematics, Wright State University

Improved apriori bounds for solutions of singularly perturbed boundary value problems are developed. Boundary layers are detected through the exponential terms in the apriori bounds. On the boundary layer domain, a singular perturbation problem is dominated by the singular perturbation parameter. On the non-boundary layer domain, such a problem is controlled by a stable reduced equation, which is independent to the perturbation. Various numerical schemes can be designed to better fit solutions of singular perturbation problems based on the boundary layer detection.

Interface-aligned Adaptive Mesh Method for Free Interface Problems Based on Level-set Formulation

Xiaoming Zheng, Mathematics, Central Michigan University

We present a novel two-dimensional interface-aligned (or body-fitting) adaptive triangular mesh method to solve free interface problems with level-set formulation. The interface-aligned mesh is achieved by two operations: projection of mesh nodes which are adjacent to the interface onto the interface, and insertion of mesh nodes right on the interface, where the combination of these two operations can effectively avoid extremely stretched triangles. The overall mesh quality is further improved by the adaptive mesh strategy, including addition/subtraction and equilibration of mesh nodes. The domain separation feature of this mesh allows the use of simpler numerical schemes compared with methods based on regular mesh, such as immersed interface method, matched interface and boundary method, etc, and second order accuracy is achieved for elliptic problems with jump conditions across the interface. Furthermore, applications to evolution of free interface problems, such as drop shear in Stokes flow and tumor growth problem are presented, and accuracies are demonstrated with comparisons with boundary integral methods.

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