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

  • Authors:

  • Khamron Mekchay;Ricardo H. Nochetto

  • Affiliations:

  • -;-

  • Venue:
  • SIAM Journal on Numerical Analysis
  • Year:
  • 2005

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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.