A Direct Method for Solving Three-Dimensional Elliptic Interface Problems

This paper presents a direct method for efficiently solving three-dimensional elliptic interface problems featuring piecewise constant coefficients with a finite jump across the interface. A key advantage of our approach lies in its avoidance of augmented variables, distinguishing it from traditional methods. The computational framework relies on a finite difference scheme implemented on a uniform Cartesian grid system. By utilizing a seven-point Laplacian for grid points away from the interface, our method only requires coefficient modifications for grid points located near or on the interface. Numerical experiments validate our method’s effectiveness. Generally, it achieves second-order accuracy for both the solution and its gradient, measured in the maximum norm, particularly effective in scenarios with moderate coefficient jumps. Extending and building upon the recent work of [1] on 1D and 2D elliptic interfaces, our approach successfully introduces a simpler method for extension into three dimensions. Notably, our proposed method not only offers efficiency and accuracy but also enhances the simplicity of implementation, making it accessible to non-experts in the field.