Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

TL;DR

This paper introduces an adaptive finite element method for nonsymmetric elliptic PDEs that combines iterative symmetrization with optimal iterative solvers, achieving linear convergence and optimal computational cost.

Contribution

It develops an inexact adaptive iteratively symmetrized FEM that ensures convergence and optimality in computational efficiency for nonsymmetric elliptic PDEs.

Findings

Proves linear convergence of the method.

Establishes optimal convergence rates with respect to computational cost.

Numerical experiments confirm theoretical results.

Abstract

We consider a general nonsymmetric second-order linear elliptic PDE in the framework of the Lax-Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive mesh-refinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, e.g., an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method (AISFEM) leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.