Computer Methods in Applied Mechanics and Engineering
A comparison of adaptive refinement techniques for elliptic problems
ACM Transactions on Mathematical Software (TOMS)
Piecewise solenoidal vector fields and the Stokes problem
SIAM Journal on Numerical Analysis
Local bisection refinement for N-simplicial grids generated by reflection
SIAM Journal on Scientific Computing
A recursive approach to local mesh refinement in two and three dimensions
Journal of Computational and Applied Mathematics
A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
Journal of Computational Physics
A Local Regularization Operator for Triangular and Quadrilateral Finite Elements
SIAM Journal on Numerical Analysis
A discontinuous hp finite element method for diffusion problems
Journal of Computational Physics
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Performance of Discontinuous Galerkin Methods for Elliptic PDEs
SIAM Journal on Scientific Computing
Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems
SIAM Journal on Numerical Analysis
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
An Adaptive Uzawa FEM for the Stokes Problem: Convergence without the Inf-Sup Condition
SIAM Journal on Numerical Analysis
Convergence of Adaptive Finite Element Methods
SIAM Review
An hp-Analysis of the Local Discontinuous Galerkin Method for Diffusion Problems
Journal of Scientific Computing
Thehp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations
Mathematics of Computation
SIAM Journal on Numerical Analysis
Adaptive Finite Element Methods with convergence rates
Numerische Mathematik
Journal of Scientific Computing
Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs
SIAM Journal on Numerical Analysis
Optimality of a Standard Adaptive Finite Element Method
Foundations of Computational Mathematics
Convergence of Adaptive Discontinuous Galerkin Approximations of Second-Order Elliptic Problems
SIAM Journal on Numerical Analysis
A Posteriori Error Estimation for Discontinuous Galerkin Finite Element Approximation
SIAM Journal on Numerical Analysis
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method
SIAM Journal on Numerical Analysis
A Multilevel Preconditioner for the Interior Penalty Discontinuous Galerkin Method
SIAM Journal on Numerical Analysis
Convergence Analysis of an Adaptive Interior Penalty Discontinuous Galerkin Method
SIAM Journal on Numerical Analysis
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We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension $\geq2$. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the ADFEM. We show that the ADFEM (and the AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.