A posteriorierror estimates for a nonconforming finite element discretization of the heat equation

Abstract

Abstract. The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in R d , d = 2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results with minimal assumptions on the mesh. Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions.Mathematics Subject Classification. 65N15, 65N30, 65M50.

A posteriori upper and lower error bound of the high-order discontinuous Galerkin method for the heat conduction equation

“…In this paper, first, we extend the results from [15] applying the approach from [12] to the high-order discontinuous Galerkin discretization. Then, we derive a lower error bound using a technique based on testing with suitable cut-off functions.…”

“…See, e.g., [12] on how to handle the right-hand side oscillation. The solution of (3.1) is called the semi-discrete solution.…”

“…We also introduce Helmholtz decomposition and an appropriate interpolation operator, as they form the basis of the presented approach. It was developed in [12], where the heat equation was solved with the aid of the combination of the CrouzeixRaviart nonconforming finite elements in space and the backward Euler scheme in time. However, the idea of using Helmholtz decomposition for splitting the error into conforming and nonconforming parts goes back to the paper [4].…”

“…We were inspired by [12], where Crouzeix-Raviart finite element method is employed for spatial discretization of the heat equation. The derived a posteriori error estimates are based on the Helmholtz decomposition of the gradient of the error.…”

A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation

“…Some other results on a posteriori error estimates for parabolic problems in a conforming setting can also be found in [24] using the so-called functional approach where a flux reconstruction is also considered, but without enforcing any local condition; furthermore, only error upper bounds are derived. Finally, we observe that contrary to conforming finite elements, a posteriori energy-norm error estimates for the heat equation discretized by nonconforming methods are less explored; we mention, in particular, [10] for mixed finite elements, [23] for nonconforming finite elements, [17] for discontinuous Galerkin methods, and [3] for finite volume schemes. This paper is organized as follows.…”