ADAPTIVE CALCULATION OF VARIABLE COEFFICIENTS
Amir Z. Averbuch, Leonid Beliak, Moshe Israeli · 2008
We propose a solver for 1-D strictly elliptic linear PDE's with non-constant coefficients of the formU −b(x)U = f(x). We combine a sparse multiplication algorithm with a diagonally preconditioned conjugate gradient (CG) method. We use sparse data structures to take advantage of the O(Ns) complexity of the algorithm, where Ns is the number of significant coefficients (i.e. above a certain threshold) required for a given accuracy. We show that the usage of a sparse multiplication in wavelet space rather than in the original physical space can speed up the performance of the sparse solver by a factor of 20. We present an algorithm and numerical results for an adaptive multiplication scheme that can rapidly solve the equation above. We explore, in detail, how the accuracy of the wavelet-based multiplication is affected by different input parameters for the algorithm. We integrated a sparse multiplication into the PDE solver. The relation between the performance of the solver and the parameters of the wavelet based sparse multiplication is also studied. This integration allowed us to extend the fast adaptive algorithms to achieve numerical solutions of linear non-constant coefficient differential equa- tions. One-dimensional numerical examples for using stand alone sparse mul- tiplication and for the differential equations solver are presented. This work, which is based on (1), extends it.