Open AccessThe boundary element method for solving the Laplace equation in two‐dimensions with oblique derivative boundary conditions
Author(s) -
Lesnic D.
Publication year - 2007
Publication title -
communications in numerical methods in engineering
Language(s) - English
Resource type - Journals
eISSN - 1099-0887
pISSN - 1069-8299
DOI - 10.1002/cnm.947
Subject(s) - singular boundary method , method of fundamental solutions , mixed boundary condition , neumann boundary condition , boundary knot method , mathematics , boundary element method , mathematical analysis , boundary (topology) , boundary value problem , poincaré–steklov operator , robin boundary condition , laplace's equation , discretization , free boundary problem , directional derivative , finite element method , physics , thermodynamics
In this communication, we extend the Neumann boundary conditions by adding a component containing the tangential derivative, hence producing oblique derivative boundary conditions. A variant of Green's formula is employed to translate the tangential derivative to the fundamental solution in the boundary element method (BEM). The two‐dimensional steady‐state heat conduction with the imposed oblique boundary condition has been tested in smooth, piecewise smooth and multiply connected domains in which the Laplace equation is the governing equation, producing results at the boundary in excellent agreement with the available analytical solutions. Convergence of the normal and tangential derivatives at the boundary is also achieved. The numerical boundary data are then used to successfully calculate the values of the solution at interior points again. The outlined test cases have been repeated with various boundary element meshes, indicating that the accuracy of the numerical results increases with increasing boundary discretization. Copyright © 2006 John Wiley & Sons, Ltd.