Mechanical quadrature method and splitting extrapolation algorithm for the boundary integral equation of the axisymmetric anisotropic Darcy's equation

ABSTRACT

In this paper, we consider the numerical solution of the three-dimensional axisymmetric anisotropic Darcy's equation with Dirichlet boundary conditions. By the potential theory and the indirect boundary element methods, we transform the system to be a one-dimensional boundary integral equation. The mechanical quadrature method (MQM) is developed to solve the integral equation. Moreover, the errors of MQM have a multivariate asymptotic expansion with the order of $O\left(h_i^3\right)$ for all mesh widths $h_i$. Once discrete equations are solved in parallel, the accuracy of numerical approximations can be improved to be the order of $O\left(h_{max}^5\right)$ by splitting extrapolation algorithm. Moreover, a posteriori asymptotic error estimate is derived to construct self-adaptive algorithms.

KEYWORDS:

• Mechanical quadrature method
• splitting extrapolation algorithm
• a posteriori estimate
• axisymmetric
• Darcy's equation

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N38
• 65R20

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11371079].