Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators

We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptivemethods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general

A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge--Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive algorithm

Finite Element Convergence for the Joule Heating Problem with Mixed Boundary Conditions

We prove strong convergence of conforming finite element approximations to the stationary Joule heating problem with mixed boundary conditions on Lipschitz domains in three spatial dimensions. We show optimal global regularity estimates on creased domains and prove a priori and a posteriori bounds for shape regular meshes.Comment: Keywords: Joule heating problem, thermistors, a posteriori error analysis, a priori error analysis, finite element metho

A posteriori error estimates for the virtual element method

An a posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing

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

Multigrid methods for nonconforming finite element methods

The present paper analyzes a multigrid algorithm for the Crouzeix-Raviart discretization of the Poisson and Stokes equations in two and three dimensions. The central point is the construction of easily computable L2-projections based on suitable quadrature rules for the transfer from coarse to fine grids and vice versa