Some Remarks on Stable Multiscale Bases for the Numerical Solution of Elliptic Partial Differential Equations

Angela Kunoth

Abstract. To be able to obtain optimally preconditioned stiffness matrices stemming from elliptic boundary value problems, one often needs stable multiscale bases on the underlying bounded domains. In particular, when appending the boundary conditions by Lagrange multipliers to uncouple them from the partial differential operator, the problem of constructing a preconditioner for the Schur complement, an operator of negative order, arises. In this context, it is very useful to explicitly know wavelet bases since then the preconditioner consists of multiplying with a linear operator performing a change of bases followed by a diagonal matrix which can be performed in O(N) arithmetic operations for N unknowns. The presentation briefly summarizes the general construction of appropriate multilevel biorthogonal bases on the interval that constitute a Riesz basis for L_{2}([0, 1]). The talk is based on the results developed in [W. Dahmen, A. Kunoth, K. Urban, Biorthogonal Spline Wavelets on the Interval – Stability and Moment Conditions, in preparation].

Angela Kunoth Weierstrass–Institut Mohrenstr. 39, 10117 Berlin, Germany kunoth@wias-berlin.de http://www.isc.tamu.edu/∼kunoth/angela.html