Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
First-order system least squares for second-order partial differential equations: part I
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
A least-squares approach based on a discrete minus one inner product for first order systems
Mathematics of Computation
Multigrid
Convergence of Adaptive Finite Element Methods
SIAM Review
Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence
SIAM Journal on Numerical Analysis
A Local A Posteriori Error Estimator Based on Equilibrated Fluxes
SIAM Journal on Numerical Analysis
A New Finite Element Gradient Recovery Method: Superconvergence Property
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements
SIAM Journal on Numerical Analysis
A Note on Constant-Free A Posteriori Error Estimates
SIAM Journal on Numerical Analysis
Recovery-Based Error Estimators for Interface Problems: Mixed and Nonconforming Finite Elements
SIAM Journal on Numerical Analysis
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In this paper, we first study two flux recovery procedures for the conforming finite element approximation to general second-order elliptic partial differential equations. One is accurate in a weighted $L^2$ norm studied in [Z. Cai and S. Zhang, SIAM J. Numer. Anal., 47 (2009), pp. 2132-2156] for linear elements, and the other is accurate in a weighted $H(\mathrm{div})$ norm, up to the accuracy of the current finite element approximation. For the $L^2$ recovered flux, we introduce and analyze an a posteriori error estimator that is more accurate than the explicit residual-based estimator. Based on the $H(\mathrm{div})$ recovered flux, we introduce two a posteriori error estimators. One estimator may be regarded as an extension of the recovery-based estimator studied in [Z. Cai and S. Zhang, SIAM J. Numer. Anal., 47 (2009), pp. 2132-2156] to higher-order conforming elements. The global reliability and the local efficiency bounds for this estimator are established provided that the underlying problem is neither convection- nor reaction-dominant. The other is proved to be exact locally and globally on any given mesh with no regularity assumptions with respect to a norm depending on the underlying problem. Numerical results on test problems for these estimators are also presented.