A 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.

中文翻译：

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一类扩散控制偏积分微分方程的分层矩阵法

提出了一种用于解决以扩散为主的偏积分微分问题的分层矩阵方法。相应的扩散控制微分算子由二阶精确有限体积方案离散，而 Fredholm 积分项由梯形规则近似。分层矩阵方法用于逼近所产生的代数问题，并包括实现高效的预处理广义最小残差 (GMRes) 求解器。这种方法通过考虑合成代数问题中以扩散为主的微分算子的固有特征，扩展了先前关于边界元方法积分形式的工作。给出了有限体积和梯形规则近似的精度和稳定性的数值分析估计，并与分层矩阵近似的估计和 GMRes 迭代的精度相结合。报告的数值实验结果成功地验证了理论精度和收敛性估计，并证明了所提出的解决方案程序的几乎最佳计算复杂度。