Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations

Abstract

We propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker-Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange finite elements of degree 1, it is locally conservative after a local postprocess giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.

A variational finite volume scheme for Wasserstein gradient flows

“…with equality on {ρ n τ > 0}. In the above formula, φ n τ denote the optimal φ realizing the sup in (19). Similarly to what has been done in the previous section for the JKO scheme, it is possible to show again that saturating inequality (20) on {ρ n τ = 0} is optimal since the mapping f → φ solution to φ + τ 2 |∇φ| 2 = f is monotone.…”

“…Problem (1) can of course be directly discretized and solved using one of the many tools available nowadays for the numerical approximation of partial differential equations. The development of energy diminishing numerical methods based on classical ODE solvers for the march in time has been the purpose of many contributions in the recent past, see for instance [8,13,16,17,19,51,56]. Nevertheless, the aforementioned methods disregard the fact that the trajectory aims at optimizing the energy decay, in opposition to methods based on minimizing movement scheme (often called JKO scheme after [34]).…”

High-order Mass-lumped Schemes for Nonlinear Degenerate Elliptic Equations

“…Coming back to the general case of (1.2) with possibly F = 0, we consider numerical methods for which the chain rule does not hold at the discrete level (as is the case for the majority of nonconforming methods). Unless the model problem is recast with a different form of nonlinearity as in [5,6], the only reasonable test function to consider in order to get estimates is v = ζ(u), which formally provides |∇ζ(u)| 2 from the diffusion term. More precisely, let us consider a scheme where the discrete unknowns z = (z i ) i∈I represent pointwise values of the solutions at certain nodes, and functions z h are reconstructed from these values and used in the weak formulation (this is the choice made in [16,1] in the case of the transient problem, through the use of "Lagrange interpolation operators").…”

Upstream mobility finite volumes for the Richards equation in heterogenous domains

“…Note that, in our theoretical development, the function p D is assumed to be defined over the whole domain Ω, which is stronger than a data p D P L 8 pΓ D q given only on the boundary. The assumption that p D does not depend on time can be removed by following the lines of [16], but we prefer here not to deal with time-dependent boundary data in order to keep the presentation as simple as possible. Finally, the initial data s 0 P L 8 pΩ; r0, 1sq in (1.1f) is also a given data.…”

“…Figure16. Water saturation profile obtained in the filling test case with the van Genuchten Mualem model using Method A and B along vertical cross-sections at different times.…”