Solving the Laplace equation by meshless collocation using harmonic kernels
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Solving the Laplace equation by meshless collocation using harmonic kernels Robert Schaback
Received: 28 June 2007 / Accepted: 30 March 2008 / Published online: 24 May 2008 © The Author(s) 2008
Abstract We present a meshless technique which can be seen as an alternative to the method of fundamental solutions (MFS). It calculates homogeneous solutions of the Laplacian (i.e. harmonic functions) for given boundary data by a direct collocation technique on the boundary using kernels which are harmonic in two variables. In contrast to the MFS, there is no artificial boundary needed, and there is a fairly general and complete error analysis using standard techniques from meshless methods for the recovery of functions. We present two explicit examples of harmonic kernels, a mathematical analysis providing error bounds and convergence rates, and some illustrating numerical examples. Keywords Laplace equation · Meshless collocation · Harmonic kernels
1 Introduction The method of fundamental solutions (MFS) solves a homogeneous boundary value problem, for example a Dirichlet problem −u = 0 in u = f in ∂
Communicated by Yuesheng Xu. R. Schaback (B) Institut fuer Numerische und Angewandte Mathematik (NAM), Georg-August-Universität Göttingen, Lotzestrasse 16-18, 37073, Göttingen, Germany e-mail: [email protected]
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R. Schaback
for the Laplace equation via approximation of the boundary data by traces of fundamental solutions centered at source points outside the domain in question. The method has been used extensively in recent years, and there are excellent surveys [1–3]. However, the method has two drawbacks: 1. It lacks a general error analysis, since the existing mathematical results are confined to concentric circles as true and “fictitious” boundaries, 2. It needs source points outside the domain which are not easy to place properly. This contribution proceeds differently by recurring to standard kernel-based reconstruction of functions from scattered data. We consider a domain given in boundary-parameterized polar form ∂ := x ∈ IR2 : x = R(t), t ∈ [0, 2π ]
R : [0, 2π ] → IR, 2π -periodic curve, R(t) = ρ(t)(cos(t), sin(t)), t ∈ [0, 2π ] and we assume that the domain is bounded by 0 < ρ(t) ≤ R < ∞ for all t ∈ [0, 2π ]. Furthermore, symmetric and positive definite harmonic kernels are constructed on IR2 × IR2 . If K is such a kernel, there are harmonic trial functions given by s(x) :=
N
a j K(x, x j), x ∈
(1)
j=1
for any set X := {x1 , . . . , x N } of N pairwise distinct points x j = R t j = ρ t j cos t j , sin t j , t j ∈ [0, 2π ], 1 ≤ j ≤ N, on the boundary ∂ of and arbitrary vectors a = (a1 , . . . , a N )T ∈ IR N . If the kernel is positive definite (see e.g. [8] for details on kernels and their applications), one can collocate a given function f : ∂ → IR on the boundary points by solving the system N
a j K xk , x j = f (xk ), 1 ≤ k ≤ N.
j=1
Then one evaluates the boundary error and uses the maximum principle to have an error bound f − s∞, ≤
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