Multigrid Solution of an Elliptic Fredholm Partial Integro-Differential Equation with a Hilbert-Schmidt Integral Operator

Gathungu, Duncan; Borzì, Alfio

doi:10.4236/am.2017.87076

Cited by 5 publications

(1 citation statement)

References 25 publications

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“…For the first term, we have the estimate of Theorem 2. The second term is estimated by (13). The third term is estimated by (14) and the convergence property of the preconditioned GMRes algorithm.…”

Section: Analysis Of the Pide Discretizationmentioning

confidence: 99%

Hierarchical‐matrix method for a class of diffusion‐dominated partial integro‐differential equations

Gathungu

Bebendorf

Borzì

2021

Numerical Linear Algebra App

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A hierarchical matrix approach for solving diffusion-dominated partial integro-differential problems is presented. The corresponding diffusiondominated differential operator is discretized by a second-order accurate finite-volume scheme, while the Fredholm integral term is approximated by the trapezoidal rule. The hierarchical matrix approach is used to approximate the resulting algebraic problem and includes the implementation of an efficient preconditioned generalized minimum residue (GMRes) solver. This approach extends previous work on integral forms of boundary element methods by taking into account inherent characteristics of the diffusion-dominated differential operator in the resultant algebraic problem. Numerical analysis estimates of the accuracy and stability of the finite-volume and the trapezoidal rule approximation are presented and combined with estimates of the hierarchical-matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical accuracy and convergence estimates, and demonstrate the almost optimal computational complexity of the proposed solution procedure.

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Section: Analysis Of the Pide Discretizationmentioning

confidence: 99%