Multigrid Analysis of Finite Element Methods with Numerical Integration

We analyze multigrid convergence rates when elliptic boundary value problems are discretized using finite element methods with numerical integration. The resulting discrete problem does not fall into the standard variational framework for analyzing multigrid methods since the bilinear forms on different grid levels are not suitably related to each other. We first discuss extensions of the standard variational multigrid theory and then apply these results to the case of numerical quadrature. In particular, it is shown that the $\mathscr{V}$-cycle algorithm has a convergence rate independent of grid size under suitable conditions.