We make a comparison of the finite difference and multiresolution method for solving the elliptic equations on irregular domains. The Dirichlet boundary condition is treated by the ghost fluid method (GFM) for the finite difference method and the Lagrange multiplier for the multiresolution method. Numerical results illustrate the improved convergence rate of errors and their gradients with the multiresolution method, up to eighth-order. Moreover, the wavelet-based multiresolution has an advantage of preconditioning of the linear system arising from the discretization of elliptic equation over the finite difference method.

中文翻译：

---

不规则域上具有Dirichlet边界条件的椭圆型方程的有限差分和多分辨率方法的比较

我们比较了求解不规则域上椭圆方程的有限差分和多分辨率方法。Dirichlet边界条件用幽灵流体法（GFM）进行有限差分法处理，而Lagrange乘数用于多分辨率方法。数值结果表明，使用多分辨率方法可以提高误差及其梯度的收敛速度，最高可达八阶。此外，与有限差分法相比，基于小波的多分辨率具有对椭圆方程离散化所产生的线性系统进行预处理的优势。