FEM Convergence for PDEs with Point Sources in 2-D and 3-D

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graf_presentation.pdf (451.24 KB)

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https://www.comsol.com/paper/fem-convergence-for-pdes-with-point-sources-in-2-d-and-3-d-26022

Permanent Link

http://hdl.handle.net/11603/11553

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UMBC Mechanical Engineering Department
UMBC Faculty Collection
UMBC Mathematics and Statistics Department
UMBC Student Collection

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Author/Creator ORCID

Date

2015

Type of Work

Technical Report
Text

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Citation of Original Publication

Gobbert, M., Kalayeh, K. M., Graf, J. S. FEM Convergence for PDEs with Point Sources in 2-D and 3-D. Presented at COMSOL Conference 2015, Boston, MA..

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Subjects

Poisson equation
Point Source
Dirac delta distribution
convergence study
mesh re finement
UMBC High Performance Computing Facility (HPCF)

Abstract

Numerical theory provides the basis for quantification of the accuracy and reliability of a FEM solution by error estimates on the FEM error vs. the mesh spacing of the FEM mesh. This paper presents techniques needed in COMSOL 5.1 to perform computational studies for elliptic test problems in two and three space dimensions that demonstrate this theory by computing the convergence order of the FEM error. In particular, we show how to perform these techniques for a problem involving a point source modeled by a Dirac delta distribution as forcing term. This demonstrates that PDE problems with a non-smooth source term necessarily have degraded convergence order compared to problems with smooth right-hand sides and thus can be most efficiently solved by low-order FEM such as linear Lagrange elements.