A locally modified second-order finite element method for interface problems

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Date
2020
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Judakova, Gozel
Richter, Thomas
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Abstract
The locally modified finite element method, which is introduced in [Frei, Richter: SINUM 52(2014), p. 2315-2334] is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the L2-norm and in a modified energy norm, as well as a reduced convergence order of O(h3/2) in the standard H1-norm. Finally, we present numerical examples to substantiate the theoretical findings.
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510 Mathematics
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ISO 690FREI, Stefan, Gozel JUDAKOVA, Thomas RICHTER, 2020. A locally modified second-order finite element method for interface problems
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@unpublished{Frei2020-07-27T23:08:48Zlocal-55628,
  year={2020},
  title={A locally modified second-order finite element method for interface problems},
  author={Frei, Stefan and Judakova, Gozel and Richter, Thomas}
}
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    <dcterms:abstract xml:lang="eng">The locally modified finite element method, which is introduced in [Frei, Richter: SINUM 52(2014), p. 2315-2334] is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the L&lt;sup&gt;2&lt;/sup&gt;-norm and in a modified energy norm, as well as a reduced convergence order of O(h&lt;sup&gt;3&lt;/sup&gt;/&lt;sup&gt;2&lt;/sup&gt;) in the standard H&lt;sup&gt;1&lt;/sup&gt;-norm. Finally, we present numerical examples to substantiate the theoretical findings.</dcterms:abstract>
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