Existence Of Discrete Solutions Research Articles

Structure-preserving semi-convex-splitting numerical scheme for a Cahn–Hilliard cross-diffusion system in lymphangiogenesis

A fully discrete semi-convex-splitting finite-element scheme with stabilization for a Cahn–Hilliard cross-diffusion system is analyzed. The system consists of parabolic fourth-order equations for the volume fraction of the fiber phase and solute concentration, modeling pre-patterning of lymphatic vessel morphology. The existence of discrete solutions is proved, and it is shown that the numerical scheme is energy stable up to stabilization, conserves the solute mass, and preserves the lower and upper bounds of the fiber phase fraction. Numerical experiments in two space dimensions using FreeFem illustrate the phase segregation and pattern formation.

Numerical method for two-dimensional linearly elastic clamped plate model

ABSTRACT In this paper, we propose an efficient numerical method for the two-dimensional (2D) linearly elastic clamped plate model. Specifically, we choose the linear Lagrange element to discretize the first two components in the vector displacement space, as well as the Morley element to discretize the third component. Moreover, the existence of discrete solution, uniqueness and a-priori error estimate is analysed. Finally, numerical experiments are presented to verify theoretical results.

A discrete boundedness-by-entropy method for finite-volume approximations of cross-diffusion systems

Abstract An implicit Euler finite-volume scheme for general cross-diffusion systems with volume-filling constraints is proposed and analyzed. The diffusion matrix may be nonsymmetric and not positive semidefinite, but the diffusion system is assumed to possess a formal gradient-flow structure that yields $L^\infty $ bounds on the continuous level. Examples include the Maxwell–Stefan systems for gas mixtures, tumor-growth models and systems for the fabrication of thin-film solar cells. The proposed numerical scheme preserves the structure of the continuous equations, namely the entropy dissipation inequality as well as the non-negativity of the concentrations and the volume-filling constraints. The discrete entropy structure is a consequence of a new vector-valued discrete chain rule. The existence of discrete solutions, their positivity, and the convergence of the scheme is proved. The numerical scheme is implemented for a one-dimensional Maxwell–Stefan model and a two-dimensional thin-film solar cell system. It is illustrated that the convergence rate in space is of order two and the discrete relative entropy decays exponentially.

An extremum‐preserving finite volume scheme for three‐temperature radiation diffusion equations

In this paper, an extremum‐preserving finite volume scheme is constructed for the two‐dimensional three‐temperature (2D 3‐T) radiation diffusion equations. The harmonic averaging points located at cell edge are applied to define the auxiliary unknowns, and the primary unknowns are defined at cell center. This scheme has a fixed stencil and satisfies the local conservation condition and discrete extremum principle. The existence of discrete solution is proved by using the fixed point theorem. Moreover, the stability analysis of this scheme is also presented. Numerical results illustrate that this scheme is efficient and accurate in solving the 2D 3‐T radiation diffusion equations on distorted meshes.

Two Structure-Preserving Time Discretizations for Gradient Flows

The equality between dissipation and energy drop is a structural property of gradient-flow dynamics. The classical implicit Euler scheme fails to reproduce this equality at the discrete level. We discuss two modifications of the Euler scheme satisfying an exact energy equality at the discrete level. Existence of discrete solutions and their convergence as the fineness of the partition goes to zero are discussed. Eventually, we address extensions to generalized gradient flows, GENERIC flows, and curves of maximal slope in metric spaces.

A Riemann solution approximation based on the zero diffusion–dispersion limit of Dafermos reformulation type problem

In the present work, a new numerical strategy is designed to approximate the Riemann solutions of systems of conservation laws. Here, the main difficulty comes from the definition of the discontinuous solutions. Indeed, the shock solutions are no longer selected by entropy criterion but they are defined as the zero limit of a diffusive–dispersive system. As a consequence, the solutions of interest may contain non classical shocks. In order to derive a suitable numerical approach, the Dafermos diffusion technique is adopted here. Then, the PDE initial value problem is reformulated as an ODE boundary value problem. A fourth-order finite difference scheme is introduced to approximate the solution of this ODE boundary value problem. In this work, a particular attention is paid on the existence of discrete solutions and several numerical experiments illustrate the relevance of the derived numerical strategy.

An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

AbstractWe propose an original scheme for the time discretization of a triphasic Cahn–Hilliard/Navier–Stokes model. This scheme allows an uncoupled resolution of the discrete Cahn–Hilliard and Navier‐Stokes system, which is unconditionally stable and preserves, at the discrete level, the main properties of the continuous model. The existence of discrete solutions is proved, and a convergence study is performed in the case where the densities of the three phases are the same. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq. 2013

A NUMERICAL ANALYSIS OF A REACTION–DIFFUSION SYSTEM MODELING THE DYNAMICS OF GROWTH TUMORS

We consider a reaction–diffusion system of 2 × 2 equations modeling the spread of early tumor cells. The existence of weak solutions is ensured by a classical argument of Faedo–Galerkin method. Then, we present a numerical scheme for this model based on a finite volume method. We establish the existence of discrete solutions to this scheme, and we show that it converges to a weak solution. Finally, some numerical simulations are reported with pattern formation examples.

ANALYSIS OF A MESHFREE METHOD FOR THE COMPRESSIBLE EULER EQUATIONS

Mathematical analysis is made on a mesh free method for the compressible Euler equations. In particular, the Moving Least Square Reproducing Kernel (MLSRK) method is employed for space approximation. With the backward-Euler method used for time discretization, existence of discrete solution and it's estimate are obtained under a regularity assumption of the continuous solution. The result of numerical experiment made on the biconvex airfoil is presented.

Uniform Convergence of an Exponentially Fitted Scheme for the Quantum Drift Diffusion Model

We analyze an exponentially fitted finite element scheme for the unipolar quantum drift diffusion model in one-dimensional space. The existence of discrete solutions is shown under very mild assumptions, and convergence of a subsequence is proved by compactness arguments. The scheme is constructed in such a way that it reduces in the semiclassical limit to the well-known Scharfetter--Gummel discretization for the classical drift diffusion model. We derive uniform error bounds which allow for the semiclassical limit on the discrete level. Numerical tests underlining the analytical results are presented.

Meshless method for the stationary incompressible Navier-Stokes equations

Mathematical analysis is achieved on a meshless method for the stationary incompressible Stokes and Navier-Stokes equations. In particular, the Moving Least Square Reproducing Kernel (MLSRK) method is employed. The existence of discrete solution and its error estimate are obtained. As a numerical example for convergence analysis, we compute the numerical solutions for these equations to compare with exact solutions. Also we solve the driven cavity flow numerically as a test problem.

Algorithms for Computing Motion by Mean Curvature

We propose a finite element algorithm for computing the motion of a surface moving by mean curvature. The algorithm uses the level set formulation so that changes in topology of the surface can be accommodated. Stability is deduced by showing that the discrete solutions satisfy both $L^\infty $ and $W^{1.1} $ bounds. Existence of discrete solutions and connections with Brakke flows are established. Some numerical examples and application to related problems, such as the phase field equations, are also presented.

Poisson Equation on a Sphere

Interpolation of Poisson equation by method adjusting equations to insure existence of discrete solution