Helmholtz方程边值问题奇异解的间断有限元数值方法

考虑Helmholtz方程一类边值问题奇异解的数值方法。解在边界上的奇异性来源于区域边界的角点或者混合边值问题在边界上的临界点。对这两类问题,在奇异点附近引入人工边界,利用局部齐次边界条件导出该人工边界上的一个精确的DtN边界条件,进而在奇点外围的区域上求解此边值问题。对此问题,用间断有限元求解,该方法的优点是允许网格剖分出现悬点,比经典有限元更适合自适应计算。数值结果表明算法对求解近似区域上的问题是有效的。

Discontinuous Galerkin Method for Helmholtz Boundary Value Problems with Singularities

Consider the numerical method to solve Helmholtz equation with singular boundary value.The singularity comes from the boundary reentry corner or mixed boundary value problem on the boundary of the critical point.For the two cases problems,we introduce an artificial boundary condition in the vicinity of the singular point and give an exactly DtN boundary condition using the local homogeneous boundary conditions.Thus we solve these problems in the domain without singular point.We use discontinuous Galerkin method to solve the boundary value problem.The advantages of DG are allowing the mesh has hanging nodes and adaptive computing.The numerical results show that our algorithm is effective in solving the problem in the approximate domain.