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

L∞L^\infty bounds for numerical solutions of noncoercive convection-diffusion equations

International audienceIn this work, we apply an iterative energy method à la de Giorgi in order to establish $L^\infty$ bounds for numerical solutions of noncoercive convection-diffusion equations with mixed Dirichlet-Neumann boundary conditions

On discrete functional inequalities for some finite volume schemes

We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\'e-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the space $BV(\Omega)$ into $L^{N/(N-1)}(\Omega)$ for a Lipschitz domain $\Omega \subset \mathbb{R}^{N}$, with $N \geq 2$. Finally, we give several applications to discrete duality finite volume (DDFV) schemes which are used for the approximation of nonlinear and non isotropic elliptic and parabolic problems

Discrete Sobolev--Poincaré inequalities for Voronoi finite volume approximations

We prove a discrete Sobolev-Poincare inequality for functions with arbitrary boundary values on Voronoi finite volume meshes. We use Sobolev's integral representation and estimate weakly singular integrals in the context of finite volumes. We establish the result for star shaped polyhedral domains and generalize it to the finite union of overlapping star shaped domains. In the appendix we prove a discrete Poincare inequality for space dimensions greater or equal to two

Stabilised finite element methods for non-symmetric, non-coercive and ill-posed problems. Part I: elliptic equations

In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element residual and jumps of certain derivatives of the discrete solution over element faces may be used. Under the assumption of well posedness of the partial differential equation and its associated adjoint problem we prove optimal error estimates in $H^1$ and $L^2$ norms in an abstract framework. Some examples of problems that are neither symmetric nor coercive, but that enter the abstract framework are given. First we treat indefinite convection-diffusion equations, with non-solenoidal transport velocity and either pure Dirichlet conditions or pure Neumann conditions and then a Cauchy problem for the Helmholtz operator. Some numerical illustrations are given.Comment: Second part in preparation: Stabilised finite element methods for non-symmetric, non-coercive and ill-posed problems. Part II: hyperbolic equation

An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes

International audienceWe are interested here in the numerical approximation of a family of probability measures, solution of the Chapman-Kolmogorov equation associated to some non-diffusion Markov process with Uncountable state space. Such an equation contains a transport term and another term, which implies redistribution Of the probability mass on the whole space. All implicit finite Volume scheme is proposed, which is intermediate between an upstream weighting scheme and a modified Lax-Friedrichs one. Due to the seemingly unusual probability framework, a new weak bounded variation inequality had to be developed, in order to prove the convergence of the discretised transport term. Such an inequality may be used in other contexts, such as for the study of finite Volume approximations of scalar linear or nonlinear hyperbolic equations with initial data in $L^1$. Also, due to the redistribution term, the tightness of the family of approximate probability measures had to be proven. Numerical examples are provided, showing the efficiency of the implicit finite volume scheme and its potentiality to be helpful in an industrial reliability context

A finite volume scheme for noncoercive elliptic equation with measure data

We show here the convergence of the finite volume approximate solutions of a convection-diffusion equation to a weak solution, without the usual coercitivity assumption on the elliptic operator and with weak regularity assumptions on the data. Numerical experiments are performed to obtain some rates of convergence in two and three space dimensions

A finite volume scheme for a noncoercive elliptic equation with measure data

We show here the convergence of the finite volume approximate solutions of a convection-diffusion equation to a weak solution, without the usual coercitivity assumption on the elliptic operator and withweak regularity assumptions on the data. Numerical experiments are performed to obtain some rates of convergence in two and three space dimensions