The finite element method for parabolic equations

Abstract

In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straightforwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.

A Coarsening Algorithm on Adaptive Grids by Newest Vertex Bisection and Its Applications

“…For example, we could first sort good nodes by some node-wise local error indicator and then mark part of good nodes for coarsening according certain marking strategy. For details, we refer to the manual of AFEM@matlab [6]. In this paper, we shall focus on the application of the proposed coarsening algorithm to iterative methods for solving algebraic equations.…”

“…We take an example from Chen and Jia [15] to test the performance of the proposed coarsening algorithm when applied to adaptive mesh refinement. Consider the heat equation in two A posteriori error estimations and adaptive algorithms for linear parabolic problems have been discussed by many researchers [6,7,18,19,30,28,23,15,22]. Traditionally we write a posterior error estimators in element-wise which is more convenient for marking elements with large local error for refinement.…”

Residual-Based a Posteriori Error Estimates for a Conforming Finite Element Discretization of the Stokes-Darcy Coupled Problem

“…It ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to be approximated, using an a posteriori error indicator. Ever since the pioneering work of Bieterman and Babuska [10], the adaptive finite element method based on a posteriori error estimates has been extensively investigated. In [4], two a posteriori error estimators for the mini-element discretization of the Stokes equations were presented.…”

Discretization of an Unsteady Flow Through a Porous Solid Modeled by Darcy's Equations

“…Much research work has been done on the a posteriori analysis of parabolic equations; we refer to the pioneering papers of Babuška and Bieterman [5], [6] concerning the space discretization and also to the work of Eriksson and Johnson [12], [13] and Verfürth [25], where the discretization relies on a global space-time variational formulation of the problem and the discontinuous Galerkin method applied to this formulation. Here we follow a rather different approach that uncouples as much as possible the time and space errors, according to an idea presented in [3].…”