Unstructured space-time finite element methods for optimal control of parabolic equations

TL;DR

This paper develops and analyzes unstructured space-time finite element methods for solving parabolic optimal control problems, including linear and semilinear cases, with error estimates and adaptive strategies.

Contribution

It introduces unstructured space-time finite element methods for parabolic control problems, proving well-posedness, deriving error estimates, and handling nonlinear and constrained cases.

Findings

Well-posedness of optimality systems established

Error estimates for discretization derived

Adaptive methods improve solution accuracy

Abstract

This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babu\v{s}ka's theorem, we show well-posedness of the first-order optimality systems for a typical model problem with linear state equations, but without control constraints. This is done for both continuous and discrete levels. Based on these results, we derive discretization error estimates. Then we consider a semilinear parabolic optimal control problem arising from the Schl\"ogl model. The associated nonlinear optimality system is solved by Newton's method, where a linear system, that is similar to the first-order optimality systems considered for the linear model problems, has to be solved at each Newton step. We present various numerical experiments including results for adaptive space-time finite element discretizations based on residual-type error indicators. In the last two examples, we also consider semilinear parabolic optimal control problems with box constraints imposed on the control.