Abstract
An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization of the PIDE problem with a multigrid scheme that includes a fast multilevel integration of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework.
Highlights
IntroductionA partial integro-differential equation (PIDE) is an equation composed of a partial-differential term and an integral term
An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization of the PIDE problem with a multigrid scheme that includes a fast multilevel integration of the Fredholm operator allowing the fast solution of the PIDE problem
A partial integro-differential equation (PIDE) is an equation composed of a partial-differential term and an integral term
Summary
A partial integro-differential equation (PIDE) is an equation composed of a partial-differential term and an integral term. Our approach is to combine a multigrid scheme for elliptic problems with the multigrid kernel approximation strategy developed in [15] For this purpose, we discretize our PIDE problem by finite-differences and quadrature rules and analyse the stability and accuracy of the resulting scheme in the case of A being the minus Laplace operator that is combined with a Fredholm Hilbert-Schmidt integral operator. We discretize our PIDE problem by finite-differences and quadrature rules and analyse the stability and accuracy of the resulting scheme in the case of A being the minus Laplace operator that is combined with a Fredholm Hilbert-Schmidt integral operator It is well-known that a multigrid scheme solves elliptic problems with optimal computational complexity.
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