Adaptive finite element methods for parabolic problems. I.: a linear model problem
SIAM Journal on Numerical Analysis
A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems
SIAM Journal on Numerical Analysis
Data Oscillation and Convergence of Adaptive FEM
SIAM Journal on Numerical Analysis
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
Convergence of Adaptive Finite Element Methods
SIAM Review
Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Adaptive Finite Element Methods with convergence rates
Numerische Mathematik
Journal of Scientific Computing
Journal of Scientific Computing
Convergence of Adaptive Discontinuous Galerkin Approximations of Second-Order Elliptic Problems
SIAM Journal on Numerical Analysis
Finite Elements in Analysis and Design
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 unifying theory of a posteriori error control for discontinuous Galerkin FEM
Numerische Mathematik
Journal of Computational and Applied Mathematics
SIAM Journal on Numerical Analysis
A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation
SIAM Journal on Numerical Analysis
Discontinuous Galerkin Methods: Theory, Computation and Applications
Discontinuous Galerkin Methods: Theory, Computation and Applications
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We derive energy-norm a posteriori error bounds for an Euler time-stepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case and then move to the fully discrete scheme by introducing the implicit Euler time-stepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To use this method in practice, we apply it to the interior penalty discontinuous Galerkin method, for which new a posteriori error bounds are derived. For the analysis of the time-dependent problems we use the elliptic reconstruction technique, and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it. We illustrate the theory with a series of numerical experiments aimed at (1) exploring practically the reliability and efficiency of the derived a posteriori estimates, and (2) testing them in an adaptive algorithm implementation.