二维外问题的高阶直接间断 Galerkin 与自然边界元耦合法

High Order Coupling of Direct Discontinuous Galerkin Method and Natural Boundary Element Method for Exterior Problem

研究直接间断Galerkin(DDG)与自然边界元(NBEM)耦合的方法来求解二维外无界区域问题.首先,引入圆周人工边界Γ,根据自然边界归化的原理获得Γ上DtN边界条件.然后,采用直接间断Galerkin方法求解基于Γ上Dirichlet边界条件的有界区域内部问题,再结合DtN条件获得弱变分问题.由于人工边界为圆周曲线,网络剖分后邻近圆周的单元为曲边三角形,利用曲边三角形上的迹逆估计和最佳多项式插值估计,证明了能量模下逼近解达到最优k(≥2)阶误差.数值例子说明了该方法的有效性和理论分析的正确性.

In this paper,the coupling of direct discontinuous Galerkin(DDG)and natural boundary element method(NBEM)is applied for solving exterior problem on a unbounded domain.A circular artificial boundary is introduced and DtN boundary condition is obtained according to the principle of natural boundary reduction DDG method is used to solve Poisson equation in a bounded subdomain with Dirichlet conditions on artificial boundary and weak variational problem is presented.With the trace inverse inequality and optimal polynomial approximation on curved triangles in the vicnity of circular artificial boundary,optimal kth order energy norm error estimate is proved.Some numerical results show that our method is very effective and theoretical analysis is correct.