Multigrid in H (div) and H (curl)

Abstract

We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces H(div) and H(curl ) in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products. Subject Classification (1991): 65N55, 65N30 Mathematics

Equilibrated residual error estimator for edge elements

“…We recall that the classical sequence (3.3) is exact [2,7,9,11,14]. Due to (3.6) the properties σ ∈ RT −1 and div σ = 0 imply that σ ∈ RT .…”

“…is an exact sequence [2]. This means that • the operator curl has a trivial kernel in H 1 /R; • the kernel {σ ∈ H(div) : div σ = 0} of the operator div is exactly the range of the operator curl; • the range of the operator div is exactly L 2 .…”

“…We show that the differential operators and the extended spaces still form exact sequences. The sequences generalize the de Rham sequences and the discrete analogs that were frequently used in the last years for constructing and understanding new finite element spaces [2,3,11].…”

Sparsity optimized high order finite element functions for H(div) on simplices

“…Since the matrix A s is only positive semidefinite with large non-trivial kernel space, parameter-robust preconditioning gets an issue especially for small parameters ε, as analyzed in [4] and [19]. It is shown in [35] that using solenoidal basis functions, namely (4.1) and (4.5), in the construction of the H(div)-conforming FE-basis, a parameter-robustness is provided by a parameter-robust preconditioner for the low-order space as e.g.…”

Convergence of a mixed method for a semi-stationary compressible Stokes system

“…The interpolation operators Π W h and Π V h are defined on function spaces with enough regularity to ensure that the corresponding degrees of freedom are functionals on these spaces (cf. [2,3]). This is reflected in writing W curl,2 0 ∩W 2,2 instead of merely W curl,2 , and so on.…”