15 research outputs found
A serendipity fully discrete div-div complex on polygonal meshes
In this work we address the reduction of face degrees of freedom (DOFs) for
discrete elasticity complexes. Specifically, using serendipity techniques, we
develop a reduced version of a recently introduced two-dimensional complex
arising from traces of the three-dimensional elasticity complex. The keystone
of the reduction process is a new estimate of symmetric tensor-valued
polynomial fields in terms of boundary values, completed with suitable
projections of internal values for higher degrees. We prove an extensive set of
new results for the original complex and show that the reduced complex has the
same homological and analytical properties as the original one. This paper also
contains an appendix with proofs of general Poincar\'e--Korn-type inequalities
for hybrid fields
Finite element approximation of the Levi-Civita connection and its curvature in two dimensions
We construct finite element approximations of the Levi-Civita connection and
its curvature on triangulations of oriented two-dimensional manifolds. Our
construction relies on the Regge finite elements, which are piecewise
polynomial symmetric (0,2)-tensor fields possessing single-valued
tangential-tangential components along element interfaces. When used to
discretize the Riemannian metric tensor, these piecewise polynomial tensor
fields do not possess enough regularity to define connections and curvature in
the classical sense, but we show how to make sense of these quantities in a
distributional sense. We then show that these distributional quantities
converge in certain dual Sobolev norms to their smooth counterparts under
refinement of the triangulation. We also discuss projections of the
distributional curvature and distributional connection onto piecewise
polynomial finite element spaces. We show that the relevant projection
operators commute with certain linearized differential operators, yielding a
commutative diagram of differential complexes.Comment: v2: Several revisions throughout, including major revisions to
Section 2.4, Section 5, and the beginning of Section
A finite element elasticity complex in three dimensions
A finite element elasticity complex on tetrahedral meshes is devised. The
conforming finite element is the smooth finite element developed by
Neilan for the velocity field in a discrete Stokes complex. The symmetric
div-conforming finite element is the Hu-Zhang element for stress tensors. The
construction of an -conforming finite element for symmetric
tensors is the main focus of this paper. The key tools of the construction are
the decomposition of polynomial tensor spaces and the characterization of the
trace of the operator. The polynomial elasticity complex and
Koszul elasticity complex are created to derive the decomposition of polynomial
tensor spaces. The trace of the operator is induced from a
Green's identity. Trace complexes and bubble complexes are also derived to
facilitate the construction. Our construction appears to be the first
-conforming finite elements on tetrahedral meshes without
further splits.Comment: 23 page
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
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Multigrid Methods for Hellan–Herrmann–Johnson Mixed Method of Kirchhoff Plate Bending Problems
A V-cycle multigrid method for the Hellan–Herrmann–Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity assumption. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context