Finite Volume Element Method for Solving the Elliptic Neumann Boundary Control Problems

Quanxiang Wang

doi:10.4236/am.2020.1112085

Abstract

Solving optimization problems with partial differential equations constraints is one of the most challenging problems in the context of industrial applications. In this paper, we study the finite volume element method for solving the elliptic Neumann boundary control problems. The variational discretization approach is used to deal with the control. Numerical results demonstrate that the proposed method for control is second-order accuracy in the L2 (Γ) and L∞ (Γ) norm. For state and adjoint state, optimal convergence order in the L2 (Ω) and H1 (Ω) can also be obtained.

Highlights

In this paper, we study the finite volume element method for solving the Neumann boundary control problems governed by elliptic partial differential equations
We study the finite volume element method for solving the elliptic Neumann boundary control problems
We study the finite volume element method for solving the Neumann boundary control problems governed by elliptic partial differential equations

Summary

Introduction

We study the finite volume element method for solving the Neumann boundary control problems governed by elliptic partial differential equations. Error estimates for finite element discretization of Neumann boundary control problems governed by elliptic equations are discussed in some publications. In [15] [16], the authors discussed distributed optimal control problems governed by elliptic equations by using the finite volume element methods. In [17], the authors considered the convergence analysis of discontinuous finite volume methods applied to distributed optimal control problems governed by a class of secondorder elliptic equations. We consider the finite volume element method for solving the elliptic Neumann boundary control problems.

Some Notations and Optimality Conditions

FVE Method

Numerical Examples

Example 1

Example 2

Conclusion