Optimal Error Properties of Finite Element Methods for Second Order Elliptic Dirichlet Problems

Optimal Error Properties of Finite Element Methods for Second Order Elliptic Dirichlet Problems

We use the informational approach of Traub and Wozniakowski [9] to study the variational form of the second order elliptic Dirichlet problem Lu = f on ü C RN. For /e Hr(Q), where r> -1, a quasi-uniform finite element method using n linear functional Jaf^i nas T7'(ß)-norm error 0(n~<r+1)/'v). We prove that it is asymptotically optimal among all methods using any information consisting of any n l...

A Posteriori Error Estimates for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

A residual type a posteriori error estimator is presented and analyzed for Weak Galerkin finite element methods for second order elliptic problems. The error estimator is proved to be efficient and reliable through two estimates, one from below and the other from above, in terms of an H1-equivalent norm for the exact error. Two numerical experiments are conducted to demonstrate the effectivenes...

Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems

In this paper, we present a priori error analysis for the finite element discretization of elliptic optimal control problems, where a finite dimensional control variable enters the Dirichlet boundary conditions. The analysis of finite element approximations of optimization problems governed by partial differential equations is an area of active research, see, e.g., [1, 12, 17, 18]. The consider...

A Finite Element Method for Second Order Nonvariational Elliptic Problems

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of “finite element Hessian” and a Schur complement approach to solvin...

Mortar Finite Volume Element Approximations of Second Order Elliptic Problems