Sparse approximation of FEM matrix for sheet current integro-differential equation. (English) Zbl 1215.65198
Olshevsky, Vadim (ed.) et al., Matrix methods. Theory, algorithms and applications. Dedicated to the memory of Gene Golub. Based on the 2nd international conference on matrix methods and operator equations, Moscow, Russia, July 23–27, 2007. Hackensack, NJ: World Scientific (ISBN 978-981-283-601-4/hbk). 510-522 (2010).
Summary: We consider a two-dimensional integro-differential equation for currents in thin superconducting films. The integral operator of this equation is hypersingular operator with kernel decaying as 1/R3. For numerical solution Galerkin finite element method (FEM) on triangular mesh with linear elements is used. It results in dense FEM matrix of large dimension. As the kernel is quickly decaying then off-diagonal elements of FEM matrix are small. We investigate simple sparsification approach based on dropping small entries of FEM matrix. The conclusion is that it allows to reduce to some extent memory requirements. Nevertheless for problems with large number of mesh points more complicated techniques as one of hierarchical matrices algorithms should be considered.
For the entire collection see [Zbl 1202.15006].
For the entire collection see [Zbl 1202.15006].
MSC:
| 65R20 | Numerical methods for integral equations |
| 45K05 | Integro-partial differential equations |
| 82D55 | Statistical mechanics of superconductors |
| 45E05 | Integral equations with kernels of Cauchy type |
