Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data

We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming non-physical regularity on the data. For simplicity of exposure, we mostly consider linear elliptic equations, and we briefly explain how these techniques can be adapted and extended to non-linear time-dependent meaningful models (Navier--Stokes equations, flows in porous media, etc.). These convergence techniques rely on discrete Sobolev norms and the translation to the discrete setting of functional analysis results

Unified convergence analysis of numerical schemes for a miscible displacement problem

This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a novel convergence result in $L^\infty(0,T; L^2(\Omega))$ of the approximate concentration using minimal regularity assumptions on the solution to the continuous problem. The convection term in the concentration equation is discretised using a centred scheme. We present a variety of numerical tests from the literature, as well as a novel analytical test case. The performance of two schemes are compared on these tests; both are poor in the case of variable viscosity, small diffusion and medium to small time steps. We show that upstreaming is not a good option to recover stable and accurate solutions, and we propose a correction to recover stable and accurate schemes for all time steps and all ranges of diffusion

Convergence in C([0,T];L2(Ω))C(\lbrack0,T\rbrack;L^2(\Omega)) of weak solutions to perturbed doubly degenerate parabolic equations

We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic $p$-Laplace equation. We show that, up to a subsequence, weak solutions of the perturbed problem converge uniformly-in-time to weak solutions of the original problem as the perturbed data approach the original data. We do not assume uniqueness or additional regularity of the solution. However, when uniqueness is known, our result demonstrates that the weak solution is uniformly temporally stable to perturbations of the data. Beginning with a proof of temporally-uniform, spatially-weak convergence, we strengthen the latter by relating the unknown to an underlying convex structure that emerges naturally from energy estimates on the solution. The double degeneracy --- shown to be equivalent to a maximal monotone operator framework --- is handled with techniques inspired by a classical monotonicity argument and a simple variant of the compensated compactness phenomenon.Comment: J. Differential Equations, 201

Numerical analysis of a two-phase flow discrete fracture matrix model

International audienceWe present a new model for two phase Darcy flows in fractured media, in which fractures are modelled as submanifolds of codimension one with respect to the surrounding domain (matrix). Fractures can act as drains or as barriers, since pressure discontinuities at the matrix-fracture interfaces are permitted. Additionally, a layer of damaged rock at the matrix-fracture interfaces is accounted for. The numerical analysis is carried out in the general framework of the Gradient Discretisation Method. Compactness techniques are used to establish convergence results for a wide range of possible numerical schemes; the existence of a solution for the two phase flow model is obtained as a byproduct of the convergence analysis. A series of numerical experiments conclude the paper, with a study of the influence of the damaged layer on the numerical solution

Analytical and numerical investigation of an intracellular calcium dynamics model

Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Dissertation, 2017von M. Sc. Jared Ouma OkiroLiteraturverzeichnis: Seite 115-12