Multigrid method for H (div) in three dimensions

Abstract

. We are concerned with the design and analysis of a multigrid algorithm for H(div; ## --elliptic linear variational problems. The discretization is based on H(div; --conforming Raviart--Thomas elements. A thorough examination of the relevant bilinear form reveals that a separate treatment of vector fields in the kernel of the divergence operator and its complement is paramount. We exploit the representation of discrete solenoidal vector fields as curls of finite element functions in so-called Nedelec spaces. It turns out that a combined nodal multilevel decomposition of both the Raviart--Thomas and Nedelec finite element spaces provides the foundation for a viable multigrid method. Its Gau--Seidel smoother involves an extra stage where solenoidal error components are tackled. By means of elaborate duality techniques we can show the asymptotic optimality in the case of uniform refinement. Numerical experiments confirm that the typical multigrid efficiency is actually achieved for model...