Abstract

A new type of meshless method is proposed in this paper to solve regularized Laplacian boundary value problems of the form $\mathcal{L}u:=r^{2}u-\Delta u\equiv 0.$ In this method, solving the Laplacian boundary value problem is starting from finding regularized Steklov eigenpairs which could provide the orthonormal basis of the space of solutions. The solutions are represented by orthogonal regularized Steklov eigenfunctions. Error bounds of Steklov approximations are provided in terms of the traces spaces of solutions. When rectangular domains are considered, explicit formulas for the regularized Steklov eigenpairs are provided. Some numerical experiments are presented to support the efficiency and accuracy of the new method.

摘要

提出了一种新型的无网格方法来解决形式为正则的拉普拉斯边值问题 $\mathcal{大号}ü：={\mathrm{[R}}^2ü-\Delta ü\equiv 0。$在这种方法中，拉普拉斯边值问题的解决是从找到正规化的Steklov特征对开始的，这可以提供解空间的正交基础。解由正交正则化Steklov特征函数表示。根据解的迹线空间提供了Steklov近似的误差范围。当考虑矩形域时，提供了正则化Steklov特征对的显式。提出了一些数值实验，以支持新方法的效率和准确性。