We generalize the a posteriori techniques for the linear heat equation in [Ver03] to the case of the nonlinear parabolic p-Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression. This leads to equivalence of th...

In this paper, we develop functional a posteriori error estimates for DG approximations of elliptic boundary-value problems. These estimates are based on a certain projection of DG approximations to the respective energy space and functional a posteriori estimates for conforming approximations (see [30, 31]). On these grounds we derive two-sided guaranteed and computable bounds for the errors i...

We present a robust posteriori error estimator for weak Galerkin finite element method applied to stationary convection-diffusion equations in the convection-dominated regime. The provides global upper and lower bounds of is sense that are uniformly bounded with respect diffusion coefficient. we use was developed by Lin, Ye, Zhang, Zhu (2018) problem without assuming any additional conditions o...

A posteriori error estimators based on quasi-norm gradient recovery are established for the finite element approximation of the p-Laplacian on unstructured meshes. The new a posteriori error estimators provide both upper and lower bounds in the quasi-norm for the discretization error. The main tools for the proofs of reliability are approximation error estimates for a local approximation operat...

In this article, an adaptive least-squares mixed finite element method is studied for pseudo-parabolic integro-differential equations. The solutions of least-squares mixed weak formulation and mixed finite element are proved. A posteriori error estimator is constructed based on the least-squares functional and the posteriori errors are obtained. Keywords—Pseudo-parabolic integro-differential eq...

This paper presents a posteriori error estimates for Signorini’s problem which is discretized via a mixed finite element approach. The error control relies on the estimation of the discretization error of an auxiliary problem given as a variational equation. The resulting error estimates capture the discretization error of the auxiliary problem, the geometrical error and the error given by the ...

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

In this paper we present an algorithm for adaptive sparse grid approximations of quantities of interest computed from discretized partial differential equations. We use adjoint-based a posteriori error estimates of the physical discretization error and the interpolation error in the sparse grid to enhance the sparse grid approximation and to drive adaptivity of the sparse grid. Utilizing these ...