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

Abstract

AbstractA hierarchical matrix approach for solving diffusion‐dominated partial integro‐differential problems is presented. The corresponding diffusion‐dominated 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.