Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

Abstract EN

We present a new second-order accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries.

Nontraditional finite element method with non-body-fitting grids is implemented on a fictitious domain in which the irregular domains are embedded.

First we set the function and coefficient in the fictitious part, and the nonsmooth boundary is then treated as an interface.

The emphasis is on the construction of jump conditions on the interface; a special position for the ghost point is chosen so that the method is more accurate.

The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear satisfying the jump conditions across the interface.

This is an efficient method for dealing with elliptic equations in irregular domains with non-smooth boundaries, and it is able to treat the general case of matrix coefficient.

The complexity and computational expense in mesh generation is highly decreased, especially for moving boundaries, while robustness, efficiency, and accuracy are promised.

Extensive numerical experiments indicate that this method is second-order accurate in the L∞ norm in both two and three dimensions and numerically very stable.