Wavelet-Galerkin methods: An adapted biorthogonal wavelet basis
The objective of this paper is the construction of a biorthogonal wavelet basis (a function \(f\) is called a wavelet if the scaled (by \(2^ j\), \(j\in Z)\) and integer translated versions of \(f\) form an orthonormal basis of \(L^ 2(\mathbb{R}))\) adapted to a given differential operator. The authors consider some one-dimensional model problems (leading to simple ODEs) to explore the principle possibilities of the adaptation of wavelets to a given problem. A multiresolution analysis approximation such that the resulting wavelets are adapted to simple model problems of the form \(u^{(2m)}(x)=g(x)\) is constructed. It is shown that even for more general elliptic differential operators the condition number of the stiffness matrix is independent of the refinement level.
condition number of the stiffness matrix
scientific article; zbMATH DE number 205300
scientific article