Monotone difference schemes for weakly coupled elliptic and parabolic systems

The present paper is devoted to the development of the theory of monotone difference schemes, approximating the so-called weakly coupled system of linear elliptic and quasilinear parabolic equations. Similarly to the scalar case, the canonical form of the vector-difference schemes is introduced and the definition of its monotonicity is given. This definition is closely associated with the property of non-negativity of the solution. Under the fulfillment of the positivity condition of the coefficients, two-side estimates of the approximate solution of these vector-difference equations are established and the important a priori estimate in the uniform norm C is given

Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure

We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure, allowing the use of some nonlinear test functions in the analysis. The existence of a solution to and the convergence of the scheme are proved under very general assumptions on the continuous problem (nonlinearities, anisotropy, heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the efficiency and of the robustness of our approach

Finite volume schemes and Lax-Wendroff consistency

We present a (partial) historical summary of the mathematical analysis of finite differences and finite volumes methods, paying a special attention to the Lax-Richtmyer and Lax-Wendroff theorems. We then state a Lax-Wendroff consistency result for convection operators on staggered grids (often used in fluid flow simulations), which illustrates a recent generalization of the flux consistency notion designed to cope with general discrete functions

Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very efficient discretization schemes in a Model Predictive Controller, where only a part of the solution needs to be computed accurately. We showcase the theoretical results by means of two examples with a nonlinear heat equation on a two-dimensional domain.Comment: 29 pages, 4 figure

Mini-Workshop: Finite Elements and Layer Adapted Meshes

[no abstract available

Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.Comment: 24 pages, 3 figure

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier--thickener model illustrate the efficiency of this method

Recovery methods for evolution and nonlinear problems

Functions in finite dimensional spaces are, in general, not smooth enough to be differentiable in the classical sense and “recovered” versions of their first and second derivatives must be sought for certain applications. In this work we make use of recovered derivatives for applications in finite element schemes for two different purposes. We thus split this Thesis into two distinct parts. In the first part we derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error for fully discrete schemes of the linear heat equation. To our knowledge this is the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique introduced as an aposteriori analog to the elliptic (Ritz) projection. Our theoretical results are backed up with extensive numerical experimentation aimed at (1) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (2) deriving an adaptive method based on our estimators. An extra novelty is an implementation of a coarsening error “preindicator”, with a complete implementation guide in ALBERTA (versions 1.0–2.0). In the second part of this Thesis we propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galërkin type using conforming finite elements and applied directly to the nonvariational(or nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of the “finite element Hessian” based on a Hessian recovery and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on linear PDEs in nonvariational form. We then use the nonvariational finite element method to build a numerical method for fully nonlinear elliptic equations. We linearise the problem via Newton’s method resulting in a sequence of nonvariational elliptic problems which are then approximated with the nonvariational finite element method. This method is applicable to general fully nonlinear PDEs who admit a unique solution without constraint. We also study fully nonlinear PDEs when they are only uniformly elliptic on a certain class of functions. We construct a numerical method for the Monge–Ampère equation based on using “finite element convexity” as a constraint for the aforementioned nonvariational finite element method. This method is backed up with numerical experimentation