Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions
Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions
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DOI：
10.1016/s0045-7825(96)00007-2
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发表时间：
1997-09
影响因子：
7.2
通讯作者：
P. Krysl;T. Belytschko
中科院分区：
文献类型：
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作者：
P. Krysl;T. Belytschko
我们用无网格Galerkin方法(EFG)研究了二阶椭圆型偏微分方程组(如拉普拉斯方程或线弹性力学)在二维非凸域中的数值解。这是一种利用紧支承权函数构造形函数的无网格法。对于非凸域，使用两种方法来确定节点是否影响特定点的近似值，即包含路径准则和可见性准则。证明了对于非凸域，可见性准则导致权函数和形函数不连续。由此得到的近似不再是一致的，其收敛必须通过检查所谓的一致性项来建立。我们证明了采用不连续形函数的无网格伽辽金方法的变种是收敛的，并且在实际重要的线性形函数情况下，收敛速度不受不连续形函数的影响。首先利用经典的和广义的斑块检验证明了间断逼近的收敛。由于这些检验不提供收敛速度的估计，因此通过直接检查误差项来检查连续和不连续的EFG形函数以及光滑和非光滑解的能量范数的收敛速度。
We consider numerical solutions of second-order elliptic partial differential equations, such as Laplace's equation, or linear elasticity, in two-dimensional, non-convex domains by the element-free Galerkin method (EFG). This is a meshless method in which the shape functions are constructed by using weight functions of compact support. For non-convex domains, two approaches to the determination of whether a node affects approximation at a particular point are used, a contained path criterion, and the visibility criterion. We show that for non-convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions. The resulting approximation is no longer conforming, and its convergence must be established by inspection of the so-called consistency term. We show that the variant of the element-free Galerkin method which uses the discontinuous shape functions, is convergent, and that, in the practically important case of linear shape functions, the convergence rate is not affected by the discontinuities. The convergence of the discontinuous approximation is first established by the classical and generalized patch test. As these tests do not provide an estimate of the convergence rate, the rate of convergence in the energy norm is examined, for both the continuous and discontinuous EFG shape functions and for smooth and non-smooth solutions by a direct inspection of the error terms.