Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
A discontinuous hp finite element method for diffusion problems
Journal of Computational Physics
Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods
SIAM Journal on Numerical Analysis
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
Error indicators for the mortar finite element discretization of the Laplace equation
Mathematics of Computation
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
A Posteriori Error Estimation for Discontinuous Galerkin Finite Element Approximation
SIAM Journal on Numerical Analysis
Discontinuous Galerkin Methods For Solving Elliptic And parabolic Equations: Theory and Implementation
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements
SIAM Journal on Numerical Analysis
Recovery-Based Error Estimators for Interface Problems: Mixed and Nonconforming Finite Elements
SIAM Journal on Numerical Analysis
Robust Equilibrated Residual Error Estimator for Diffusion Problems: Conforming Elements
SIAM Journal on Numerical Analysis
Weak Galerkin methods for second order elliptic interface problems
Journal of Computational Physics
Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes
Journal of Computational and Applied Mathematics
A posteriori error analysis for discontinuous finite volume methods of elliptic interface problems
Journal of Computational and Applied Mathematics
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Discontinuous Galerkin (DG) finite element methods were studied by many researchers for second-order elliptic partial differential equations, and a priori error estimates were established when the solution of the underlying problem is piecewise $H^{3/2+\epsilon}$ smooth with $\epsilon0$. However, elliptic interface problems with intersecting interfaces do not possess such a smoothness. In this paper, we establish a quasi-optimal a priori error estimate for interface problems whose solutions are only $H^{1+\alpha}$ smooth with $\alpha\in(0,1)$ and, hence, fill a theoretical gap of the DG method for elliptic problems with low regularity. The second part of the paper deals with the design and analysis of robust residual- and recovery-based a posteriori error estimators. Theoretically, we show that the residual and recovery estimators studied in this paper are robust with respect to the DG norm, i.e., their reliability and efficiency bounds do not depend on the jump, provided that the distribution of coefficients is locally quasi-monotone.