| Issue |
ESAIM: M2AN
Volume 57, Number 5, September-October 2023
|
|
|---|---|---|
| Page(s) | 3091 - 3111 | |
| DOI | https://doi.org/10.1051/m2an/2023076 | |
| Published online | 25 October 2023 | |
Fully discrete pointwise smoothing error estimates for measure valued initial data
1
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
2
Chair of Optimal Control, Technical University of Munich, School of Computation Information and Technology, Department of Mathematics, Boltzmannstraße 3, 85748 Garching b. Munich, Germany
3
Chair of Optimal Control, Technical University of Munich, School of Computation Information and Technology, Department of Mathematics, Boltzmannstraße 3, 85748 Garching b. Munich, Germany
* Corresponding author: vexler@ma.tum.de
Received:
3
April
2023
Accepted:
4
September
2023
In this paper we analyze a homogeneous parabolic problem with initial data in the space of regular Borel measures. The problem is discretized in time with a discontinuous Galerkin scheme of arbitrary degree and in space with continuous finite elements of orders one or two. We show parabolic smoothing results for the continuous, semidiscrete and fully discrete problems. Our main results are interior L∞ error estimates for the evaluation at the endtime, in cases where the initial data is supported in a subdomain. In order to obtain these, we additionally show interior L∞ error estimates for L2 initial data and quadratic finite elements, which extends the corresponding result previously established by the authors for linear finite elements.
Mathematics Subject Classification: 65N30 / 65N15 / 65N30 / 65N15
Key words: Optimal control / sparse control / initial data identification / smoothing estimates / parabolic problems / finite elements / discontinuous Galerkin / error estimates / pointwise error estimates
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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