A new numerical approach for the time independent Helmholtz equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. At similar 9-point stencils, the accuracy of the new approach is two orders higher for the Dirichlet boundary conditions and one order higher for the Neumann boundary conditions than that for the linear finite elements. The numerical results for irregular domains also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. The new approach can be equally applied to the Helmholtz and screened Poisson equations.

中文翻译：

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在不规则域和笛卡尔网格上以最佳精度求解二维亥姆霍兹方程的新数值方法

不规则域上时间无关 Helmholtz 方程的新数值方法已经开发出来。普通笛卡尔网格和具有未知系数的简单 9 点模板方程用于二维不规则域。模板方程系数的计算基于模板方程的局部截断误差的最小化，并产生最佳的精度顺序。在类似的 9 点模板上，新方法的精度比线性有限元的 Dirichlet 边界条件高 2 个数量级，Neumann 边界条件的精度高 1 个数量级。不规则域的数值结果还表明，在相同数量的自由度下，新方法比具有更宽模板的二次和三次有限元更准确。