A new error analysis for discontinuous finite element methods for linear elliptic problems

DOI10.1090/S0025-5718-10-02360-4zbMath1201.65198MaRDI QIDQ3160735

Thirupathi Gudi

Publication date: 8 October 2010

Published in: Mathematics of Computation (Search for Journal in Brave)

zbMATH Keywords

finite element methoddiscontinuous Galerkin methodboundary value problemoptimal error estimateselliptic regularitynonconforminglinear elliptic problemssecond and fourth order elliptic problems

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Boundary value problems for higher-order elliptic equations (35J40) Error bounds for boundary value problems involving PDEs (65N15) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains
• Two-level non-overlapping Schwarz preconditioners for a discontinuous Galerkin approximation of the biharmonic equation
• Mixed discontinuous Galerkin finite element method for the biharmonic equation
• A weakly over-penalized symmetric interior penalty method
• \(hp\)-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation
• A unifying theory of a posteriori error control for discontinuous Galerkin FEM
• A \(W\)-cycle algorithm for a weakly over-penalized interior penalty method
• A posteriori error estimation and adaptive mesh-refinement techniques
• Discrete Sobolev and Poincaré inequalities for piecewise polynomial functions
• Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity
• An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
• An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems
• Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition
• ENERGY NORM A POSTERIORI ERROR ESTIMATION OF hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS
• An a posteriori error estimator for a quadratic C0-interior penalty method for the biharmonic problem
• An Interior Penalty Finite Element Method with Discontinuous Elements
• Conforming and nonconforming finite element methods for solving the stationary Stokes equations I
• Finite Element Methods for Elliptic Equations Using Nonconforming Elements
• An Elliptic Collocation-Finite Element Method with Interior Penalties
• Convergence of nonconforming multigrid methods without full elliptic regularity
• The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
• Poincaré--Friedrichs Inequalities for Piecewise H¹ Functions
• A Posteriori Error Estimates for a Discontinuous Galerkin Approximation of Second-Order Elliptic Problems
• A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems
• Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
• Poincaré–Friedrichs Inequalities for PiecewiseH²Functions
• Two-level additive Schwarz preconditioners for nonconforming finite element methods
• The Mathematical Theory of Finite Element Methods

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