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Numerical solution of parabolic Cauchy problems in planar corner domains

Authors
Journal
Mathematics and Computers in Simulation
0378-4754
Publisher
Elsevier
Volume
101
Identifiers
DOI: 10.1016/j.matcom.2014.03.001
Keywords
  • Heat Equation
  • Cauchy Problem
  • Landweber Method
  • Mixed Boundary Value Problems
  • Corner Singularities
  • Boundary Integral Equation
Disciplines
  • Mathematics

Abstract

Abstract An iterative method for the parabolic Cauchy problem in planar domains having a finite number of corners is implemented based on boundary integral equations. At each iteration, mixed well-posed problems are solved for the same parabolic operator. The presence of corner points renders singularities of the solutions to these mixed problems, and this is handled with the use of weight functions together with, in the numerical implementation, mesh grading near the corners. The mixed problems are reformulated in terms of boundary integrals obtained via discretization of the time-derivative to obtain an elliptic system of partial differential equations. To numerically solve these integral equations a Nyström method with super-algebraic convergence order is employed. Numerical results are presented showing the feasibility of the proposed approach.

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