基于粘接元技术的区域分裂解法及其应用

Studying on domain decomposition using mortar elements and its applications

针对线性椭圆型问题 ,发展了一种无重叠区域分裂方法。首先引入Lagrange乘子 ,以使子域交界处解的连续性约束条件获得弱满足。接着重点讨论了Lagrange乘子的近似空间 ,即粘接元 (mortarelements)空间的建立 ,以及所引起的离散鞍点问题的共轭梯度迭代解法。这种区域分裂方法非常适用于网格不匹配情形。然后 ,对基于二阶鼓包 (bump)函数近似的当地事后误差估算方法提出改进 ,以使之适应于区域分裂情形。我们还将上述方法应用于广义Stokes问题。最后 ,给出了以误差估算值为准则的自适应网格上的数值结果。

In this paper, we develop a method of non overlapping domain decomposition for linear elliptic problem. First, Lagrange multiplier is introduced so that the continuity of solution at the interface between sub domains is satisfied weakly. Then we concentrate on the construction of the discrete space of Lagrange multiplier, the space of mortar elements, and the conjugated gradient method for the related saddle point problem. This domain decomposition method is very suitable for the non matching meshes. Some modifications to the local a posteriori error estimation based on the approximation with quadratic bump functions are made for domain decomposition. This method can be extended to the generalized Stokes problem. Finally, some numerical experiments were presented on the adapted meshes with a posteriori error estimate as criterion.