Multiwavelet Collocation Boundary Element Solution of Laplace's Equation

Amini, S; Elmazuzi, R; Nixon, S P

Abstract

In this paper we present the result of a preliminary investigation into the collocation method, based on multiwavelet approximation, for boundary integral equations with analytically standard kernels. We apply the method to the boundary integral solution of Laplace's equation exterior to a closed curve. We present some numerical results which show a good level of compression. Multiwavelets on [0,1] Wavelets are generated by a mother wavelet ψ(x) from which all other wavelets are obtained using the definition ψ λ (x) := 2 m 2 ψ(2 m x − l) ∀ λ := {m, l} : m, l ∈ Z, see [1]. They have proved to be efficient and effective bases for function approximations, as the coefficients of a wavelet expansion decay rapidly for a large class of functions. Due to the multiresolution property of wavelets they provide accurate local descriptions of functions efficiently. For example in the presence of corners and edges, the functions can still be approximated with a linear combination of just a few wavelet bases. Wavelets are attractive for the numerical solution of integral equations, because their vanishing moments property leads to operator compression [2], [3]. However, to obtain wavelets with compact support and high order of vanishing moments, the length of the support increases as the order of the vanishing moments increases. This causes difficulties with the practical use of wavelets particulary at edges and corners. With multiwavelets, an increase in the order of vanishing moments is obtained not by increasing the support but by increasing the number of mother wavelets. Suppose k is a positive integer and m a non-negative integer, we define the space V k m of piecewise polynomial functions V k m := f : f | [2 −m n,2 −m (n+1)] is a polynomial of degree less than k ∀ n = 0, 1,. .. , 2 m − 1 and vanishes elsewhere. It is clear that V k 0 ⊂ V k 1 ⊂. .. ⊂ V k m ⊂. .. ⊂ L 2 [0, 1]. For m = 0, 1, 2,. . ., we define the space W k m to be the orthogonal complement of V k m in V k m+1 ; that is V k m+1 = V k m ⊕W k m. Then we have the decomposition V k m = V k 0 ⊕W k 0 ⊕W k 1 ⊕. .. ⊕W k m−1. The space V k 0 is the space of polynomials of degree less than k on the interval [0,1] and we assume {φ 1 , φ 2 ,. .. , φ k } to be a basis for it. These are known as the scaling functions. Suppose {ψ 1 , ψ 2 ,. .. , ψ k } is a basis of W k 0. Therefore, for the orthogonality condition

Citation

Amini, S., Elmazuzi, R., & Nixon, S. P. (2004). Multiwavelet Collocation Boundary Element Solution of Laplace's Equation.

• Start Date: Jul 26, 2004
• Online Publication Date: Jul 29, 2004
• Publication Date: Jul 29, 2004
• Deposit Date: Jul 18, 2024
• Publisher: Tech Science Press
• Volume: 34
• ISBN: 096570016X