Unstructured Space-Time Finite Element Methods for Optimal Control of Parabolic Equations

Ulrich Langer, Olaf Steinbach, Fredi Tröltzsch, Huidong Yang

Research output: Contribution to journalArticlepeer-review


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 Babuška'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 Schlögl model. The associated nonlinear optimality system is solved by Newton's method, where a linear system, which 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.

Original languageEnglish
Pages (from-to)A744–A771
Number of pages28
JournalSIAM Journal on Scientific Computing
Issue number2
Publication statusPublished - Mar 2021


  • Discretization error estimates
  • Linear parabolic equations
  • Parabolic optimal control problems
  • Semilinear parabolic equations
  • Space-time finite element methods

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Fields of Expertise

  • Information, Communication & Computing

Cite this