A Highly-Accurate Finite Element Method with Exponentially Compressed Meshes for the Solution of the Dirichlet Problem of the generalized Helmholtz Equation with Corner Singularities

Emine Celiker (Lead / Corresponding author), Ping Lin

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1 Citation (Scopus)

Abstract

In this study, a highly-accurate, conforming finite element method is developed and justified for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. The k−th order Lagrange elements are used for the discretization of the variational form of the problem on exponentially compressed polar meshes employed in the neighbourhood of the corners whose interior angle is απ, α≠1∕2, and on the triangular and curved mesh formed in the remainder of the polygon. The exponentially compressed polar meshes are constructed such that they are transformed to square meshes using the Log-Polar transformation, simplifying the realization of the method significantly. For the error bound between the exact and the approximate solution obtained by the proposed method, an accuracy of O(h k),h mesh size and k≥1 an integer, is obtained in the H 1-norm. Numerical experiments are conducted to support the theoretical analysis made. The proposed method can be applied for dealing with the corner singularities of general nonlinear parabolic partial differential equations with semi-implicit time discretization.

Original languageEnglish
Pages (from-to)227-235
Number of pages9
JournalJournal of Computational and Applied Mathematics
Volume361
Early online date16 Apr 2019
DOIs
Publication statusPublished - 1 Dec 2019

Fingerprint

Corner Singularity
Helmholtz equation
Helmholtz Equation
Generalized Equation
Dirichlet Problem
Partial differential equations
Finite Element Method
Mesh
Finite element method
Experiments
Interior angle
Variational Form
Semi-implicit
Parabolic Partial Differential Equations
Time Discretization
Remainder
Nonlinear Partial Differential Equations
Lagrange
Error Bounds
Polygon

Keywords

  • Error analysis
  • Finite element method
  • Helmholtz equation
  • Mesh refinement
  • Singularity problem

Cite this

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title = "A Highly-Accurate Finite Element Method with Exponentially Compressed Meshes for the Solution of the Dirichlet Problem of the generalized Helmholtz Equation with Corner Singularities",
abstract = "In this study, a highly-accurate, conforming finite element method is developed and justified for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. The k−th order Lagrange elements are used for the discretization of the variational form of the problem on exponentially compressed polar meshes employed in the neighbourhood of the corners whose interior angle is απ, α≠1∕2, and on the triangular and curved mesh formed in the remainder of the polygon. The exponentially compressed polar meshes are constructed such that they are transformed to square meshes using the Log-Polar transformation, simplifying the realization of the method significantly. For the error bound between the exact and the approximate solution obtained by the proposed method, an accuracy of O(h k),h mesh size and k≥1 an integer, is obtained in the H 1-norm. Numerical experiments are conducted to support the theoretical analysis made. The proposed method can be applied for dealing with the corner singularities of general nonlinear parabolic partial differential equations with semi-implicit time discretization.",
keywords = "Error analysis, Finite element method, Helmholtz equation, Mesh refinement, Singularity problem",
author = "Emine Celiker and Ping Lin",
note = "E. Celiker's work is supported by the EU Scholarship Programme for the Turkish Cypriot Community 2017/18.",
year = "2019",
month = "12",
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AU - Celiker, Emine

AU - Lin, Ping

N1 - E. Celiker's work is supported by the EU Scholarship Programme for the Turkish Cypriot Community 2017/18.

PY - 2019/12/1

Y1 - 2019/12/1

N2 - In this study, a highly-accurate, conforming finite element method is developed and justified for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. The k−th order Lagrange elements are used for the discretization of the variational form of the problem on exponentially compressed polar meshes employed in the neighbourhood of the corners whose interior angle is απ, α≠1∕2, and on the triangular and curved mesh formed in the remainder of the polygon. The exponentially compressed polar meshes are constructed such that they are transformed to square meshes using the Log-Polar transformation, simplifying the realization of the method significantly. For the error bound between the exact and the approximate solution obtained by the proposed method, an accuracy of O(h k),h mesh size and k≥1 an integer, is obtained in the H 1-norm. Numerical experiments are conducted to support the theoretical analysis made. The proposed method can be applied for dealing with the corner singularities of general nonlinear parabolic partial differential equations with semi-implicit time discretization.

AB - In this study, a highly-accurate, conforming finite element method is developed and justified for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. The k−th order Lagrange elements are used for the discretization of the variational form of the problem on exponentially compressed polar meshes employed in the neighbourhood of the corners whose interior angle is απ, α≠1∕2, and on the triangular and curved mesh formed in the remainder of the polygon. The exponentially compressed polar meshes are constructed such that they are transformed to square meshes using the Log-Polar transformation, simplifying the realization of the method significantly. For the error bound between the exact and the approximate solution obtained by the proposed method, an accuracy of O(h k),h mesh size and k≥1 an integer, is obtained in the H 1-norm. Numerical experiments are conducted to support the theoretical analysis made. The proposed method can be applied for dealing with the corner singularities of general nonlinear parabolic partial differential equations with semi-implicit time discretization.

KW - Error analysis

KW - Finite element method

KW - Helmholtz equation

KW - Mesh refinement

KW - Singularity problem

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