This paper investigates recent progress on a-posteriori error analysis for the high-order finite element method(FEM). The paper will discuss the differences between a-posteriori error estimations for lower-order FEM and those for high-order FEM, and analyzes the technical and methodological differences on a-posteriori error estimations for high-order FEM in one dimension and in high dimensions....

In this paper, we investigate a residual-based posteriori error estimates for the hp finite element approximation of general optimal control problems governed by bilinear elliptic equations. By using the hp discontinuous Galerkin finite element approximation for the control and the hp finite element approximation for both the state and the co-state, we derive a posteriori upper error bounds for...

It is shown that, if the residual of a stable numerical solution is sufficiently small, there exists a nearby exact solution for which an a posteriori error estimate holds. This result demonstrates that it is not necessary to assume the existence of exact solutions in a posteriori error analysis for nonlinear equations. The concept is analyzed in an abstract Banach space setting and its practic...

Based on a new a posteriori error estimator, an adaptive finite element method is proposed for recovering the Robin coefficient involved in a diffusion system from some boundary measurement. The a posteriori error estimator can not be derived for this ill-posed nonlinear inverse problem as it was done for the existing a posteriori error estimators for direct problems. Instead, we shall derive t...

The adaptive finite element method based on a posteriori error estimates provides a systematic way to refine or coarsen the meshes according to the local a posteriori error estimator on the elements. One of the remarkable properties of the method is that for appropriately designed adaptive finite element procedures, the meshes and the associated numerical complexity are quasi-optimal in the sen...

In this paper, we propose and analyze a linear time stepping finite element method for abstract second order linear evolution problems. For such methods, we derive optimal order a posteriori error estimates and sharp a posteriori nodal error estimates using the energy approach and the duality argument. Based on these estimates, we further design an adaptive time stepping strategy for the previo...