Fully discrete pointwise smoothing error estimates for measure valued initial data

Dmitriy Leykekhman, Boris Vexler, Jakob Wagner

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)3091-3111
Number of pages21
JournalMathematical Modelling and Numerical Analysis
Issue number5
StatePublished - 1 Sep 2023


  • Discontinuous Galerkin
  • Error estimates
  • Finite elements
  • Initial data identification
  • Optimal control
  • Parabolic problems
  • Pointwise error estimates
  • Smoothing estimates
  • Sparse control


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