Abstract
Abstract For the modeling and numerical approximation of problems with time-dependent Dirichlet boundary conditions, one can call on several consistent and inconsistent approaches. We show that spatially discretized boundary control problems can be brought into a standard state space form accessible for standard optimization and model reduction techniques. We discuss several methods that base on standard finite element discretizations, propose a newly developed problem formulation, and investigate their performance in numerical examples. We illustrate that penalty schemes require a wise choice of the penalization parameters in particular for iterative solves of the algebraic equations. Incidentally, we confirm that standard finite element discretizations of higher order may not achieve the optimal order of convergence in the treatment of boundary forcing problems and that convergence estimates by the common method of manufactured solutions can be misleading.
Highlights
In practical applications, see [1,2] for examples on flow control, a system is typically controlled via actuations at an interface
We have made the distinction between consistent schemes and relaxed schemes that depend on a penalization parameter
Similar tests but with a volume force led to an estimated order of spatial convergence (EOC) 1⁄4 3, the quadratic elements
Summary
Published online: 6 October 2015 (version 1) Cite as: P. For the current reviewing status and the latest referee’s comments please click here or scan the QR code at the end of this article. Primary discipline: Numerical methods Secondary discipline: Numerical & Computational mathematics Keywords: Finite Element Method, Boundary Conditions, Dirichlet Boundary Conditions, Variational Formulation, Boundary Control, FEM
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