H(curl)-椭圆问题自适应内罚间断有限元方法的收敛性分析

Convergence of Adaptive Interior Penalty Discontinuous Galerkin Methos for H(curl)-Elliptic Problems

针对H(curl)-椭圆问题的自适应内罚间断有限元方法,给出了相应的收敛性证明:把间断有限元空间分解成棱有限元空间及其正交补空间,然后结合误差的整体上界估计、关于加密网格之间的网格尺寸的2个条件以及后验误差指示子的单调性等性质,证明了在连续迭代过程中,关于误差函数的能量范数与尺度化的误差指示子之和是压缩的,即自适应内罚间断有限元方法是收敛的.

The convergence of Adaptive Interior Penalty Discontinuous Galerkin methos（ AIPDG） for H（ curl）- elliptic problem is proved. To this end,the discontinuous finite element space is decomposed into the edge finite element and the corresponding orthogonal space,then the global upper bound,two conditions for the meshsizes between the refinement procedure,and the property of the error of estimator are established and combined. At last,the AIPDG is proved to be a contraction,for the sum of the norm of energy error estimation and the scaled error indicator,between two consecutive adaptive loops; namely,AIPDG is convergent.