Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs

Abstract

We prove convergence of adaptive finite element methods (AFEMs) for general (nonsymmetric) second order linear elliptic PDEs, thereby extending the result of Morin, Nochetto, and Siebert [{\it SIAM J. Numer.\ Anal.}, 38 (2000), pp. 466--488; {\it SIAM Rev.}, 44 (2002), pp. 631--658]. The proof relies on quasi-orthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEMs are a contraction for the sum of energy error plus oscillation. Numerical experiments, including oscillatory coefficients and {both coercive and noncoercive} convection-diffusion PDE, illustrate the theory and yield optimal meshes.