Refined mixed finite element method for the poisson problem in a polygonal domain with a reentrant corner

摘要

We consider the Dirichlet problem for the Laplace operator in a plane polygonal domain (simply connected) with a reentrant corner at the origin. In addition to the potential function u, the vectorfield p-Vector = (▽-Vector)u introduced as a supplementary unknown. On each triangle of a fixed triangulation, p-Vector is approximated by a Raviart-Thomas vector field of degree 0 and u by a constant function. Under suitable refinement conditions on the considered regular family of triangulations, we prove that the convergence of the sequence of approximate solutions ((p-Vector)_h, u_h) to ((p-Vector), u) is still of order 1 in the L~2-norm. The discrete problem is hybridized and the unknowns u_h and (p-Vector)_h are explicitly eliminated in terms of the Lagrange multipliers.