Vol. 10, No. 1, 2015

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
An adaptively weighted Galerkin finite element method for boundary value problems

Yifei Sun and Chad R. Westphal

Vol. 10 (2015), No. 1, 27–41
Abstract

We introduce an adaptively weighted Galerkin approach for elliptic problems where diffusion is dominated by strong convection or reaction terms. In such problems, standard Galerkin approximations can have unacceptable oscillatory behavior near boundaries unless the computational mesh is sufficiently fine. Here we show how adaptively weighting the equations within the variational problem can increase accuracy and stability of solutions on under-resolved meshes. Rather than relying on specialized finite elements or meshes, the idea here sets a flexible and robust framework where the metric of the variational formulation is adapted by an approximate solution. We give a general overview of the formulation and an algorithmic structure for choosing weight functions. Numerical examples are presented to illustrate the method.

Keywords
finite element methods, convection-dominated diffusion, boundary layers, adaptive, weighted
Mathematical Subject Classification 2010
Primary: 65N30, 65N12, 35J20
Milestones
Received: 27 October 2013
Revised: 12 June 2014
Accepted: 25 August 2014
Published: 27 March 2015
Authors
Yifei Sun
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012
United States
Chad R. Westphal
Department of Mathematics and Computer Science
Wabash College
Crawfordsville, IN 47933
United States