Complexes from Complexes

Hilbert complexes with mixed boundary conditions part 1: de Rham complex

“…9,17,18 Compact embeddings (1) corresponding to the biharmonic and the elasticity complex are given in Pauly and Zulehner 8 and their other works, 6,7 respectively. Note that in the recent paper, 1 similar results have been shown for the special case of no or full boundary conditions using an alternative and more algebraic approach, the so-called Bernstein-Gelfand-Gelfand (BGG) resolution.…”

“…are bounded and satisfy A 1  A 1 = id R(A 1 ) as well as id D(A 1 ) =  1 A 1 (1) implies several important results related to the particular Hilbert complex by the so-called FA-ToolBox, cf. previous studies [2][3][4][5] and other works.…”

Finite Element Systems for Vector Bundles: Elasticity and Curvature

“…These complexes can be deduced from vector-valued de Rham sequences by a diagram chase of the following type [10]:…”

Spurious solutions for high order curl problems

“…To explain the spurious solutions, we relate the grad curl model problem to the Hodge-Laplacian boundary value problems of the grad curl-complex, which is derived by connecting de Rham complexes in an algebraic framework inspired by the Bernstein-Gelfand-Gelfand construction [3, (46)]. As a consequence, the analytic results in [3] and tools from the finite element exterior calculus (FEEC) can be applied to the high order curl problems.…”

“…For high order curl problems, zero eigenvalues exist even on contractible domains (on the other hand, for the Maxwell equations the zero eigenvalues only exist if the domain has nontrivial topology). From the cohomological perspective, this is because the grad curlcomplex has nontrivial cohomology even on contractible domains [3] due to the high order nature of the operators. These harmonic forms correspond to zero eigenvalues.…”