Abstract.

We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is quasi-optimal in two-dimensional space and suboptimal in three-dimensional space. Numerical simulations are provided to confirm our findings.

Keywords

  1. finite elements
  2. interface problems
  3. immersed boundary method
  4. Dirac delta approximations
  5. a posteriori error estimates
  6. adaptivity

MSC codes

  1. 65N15
  2. 65N30
  3. 65N50

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Acknowledgment

The author would like to thank the anonymous reviewers who provided valuable comments and remarks on the earlier version of the manuscript.

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Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 431 - 456
ISSN (online): 1095-7170

History

Submitted: 28 October 2021
Accepted: 25 October 2022
Published online: 10 March 2023

Keywords

  1. finite elements
  2. interface problems
  3. immersed boundary method
  4. Dirac delta approximations
  5. a posteriori error estimates
  6. adaptivity

MSC codes

  1. 65N15
  2. 65N30
  3. 65N50

Authors

Affiliations

Luca Heltai
Mathematics Area, SISSA – International School for Advanced Studies, via Bonomea 265, 34136, Trieste, Italy.
School of Mathematical Sciences, University of Electronic Science and Technology of China, No2006, Xiyuan Ave, West Hi-Tech Zone, 611731, Chengdu, China.

Funding Information

Italian Ministry of Education, University, and Research
Funding: The work of the authors was partially supported by the National Research Projects (PRIN 2017) “Numerical Analysis for Full and Reduced Order Methods for the Efficient and Accurate Solution of Complex Systems Governed by Partial Differential Equations,” funded by the Italian Ministry of Education, University, and Research.

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