The divDiv-complex and applications to biharmonic equations

ABSTRACT It is shown that the first biharmonic boundary value problem on a topologically trivial domain in 3D is equivalent to three (consecutively to solve) second-order problems. This decomposition result is based on a Helmholtz-like decomposition of an involved non-standard Sobolev space of tensor fields and a proper characterization of the operator acting on this space. Similar results for biharmonic problems in 2D and their impact on the construction and analysis of finite element methods have been recently published in Krendl et al. [A decomposition result for biharmonic problems and the Hellan–Herrmann–Johnson method. Electron Trans Numer Anal. 2016;45:257–282]. The discussion of the kernel of leads to (de Rham-like) closed and exact Hilbert complexes, the -complex and its adjoint the -complex, involving spaces of trace-free and symmetric tensor fields. For these tensor fields, we show Helmholtz type decompositions and, most importantly, new compact embedding results. Almost all our results hold and are formulated for general bounded strong Lipschitz domains of arbitrary topology. There is no reasonable doubt that our results extend to strong Lipschitz domains in .

[1]  Ralf Hiptmair,et al.  Universal extension for Sobolev spaces of differential forms and applications , 2012 .

[2]  R. Picard.  Zur Theorie der harmonischen Differentialformen , 1979 .

[3]  M. E. Bogovskii.  Solution of the first boundary value problem for the equation of continuity of an incompressible medium , 1979 .

[4]  Leonard R. Herrmann,et al.  Finite-Element Bending Analysis for Plates , 1967 .

[5]  N. Weck,et al.  Time-Harmonic Maxwell Equations in the Exterior of Perfectly Conducting, Irregular Obstacles , 2001 .

[6]  Claes Johnson,et al.  On the convergence of a mixed finite-element method for plate bending problems , 1973 .

[7]  D. Pauly.  On Constants in Maxwell Inequalities for Bounded and Convex Domains , 2015 .

[8]  P. Werner,et al.  A local compactness theorem for Maxwell's equations , 1980 .

[9]  D. Arnold,et al.  Finite element exterior calculus, homological techniques, and applications , 2006, Acta Numerica.

[10]  Giovanni P. Galdi,et al.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems , 2011 .

[11]  K. Witsch,et al.  A remark on a compactness result in electromagnetic theory , 1993 .

[12]  M. Aoki.  To University of Minnesota , 2018 .

[13]  D. Pauly.  On the Maxwell constants in 3D , 2014, 1406.1723.

[14]  W. Zulehner,et al.  A Decomposition Result for Biharmonic Problems and the Hellan-herrmann-johnson Method a Decomposition Result for Biharmonic Problems and the Hellan-herrmann-johnson Method * , 2022 .

[15]  Sebastian Bauer,et al.  The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions , 2015, SIAM J. Math. Anal..

[16]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[17]  D. Pauly.  On Maxwell's and Poincare's Constants , 2013, 1311.2186.

[18]  Rolf Stenberg,et al.  A posteriori error estimates for the Morley plate bending element , 2007, Numerische Mathematik.

[19]  R. Picard.  An elementary proof for a compact imbedding result in generalized electromagnetic theory , 1984 .

[20]  Vincent Quenneville-Belair,et al.  A new approach to finite element simulations of general relativity , 2015 .

[21]  Xuehai Huang,et al.  Convergence of an Adaptive Mixed Finite Element Method for Kirchhoff Plate Bending Problems , 2011, SIAM J. Numer. Anal..

[22]  Martin Costabel,et al.  On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains , 2008, 0808.2614.

[23]  C. Simader,et al.  Direct methods in the theory of elliptic equations , 2012 .

[24]  Rainer Picard,et al.  On the boundary value problems of electro- and magnetostatics , 1982, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[25]  Frank Jochmann,et al.  A compactness result for vector fields with divergence and curl in Lq(ω) involving mixed boundary conditions , 1997 .

[26]  N. Weck,et al.  Maxwell's boundary value problem on Riemannian manifolds with nonsmooth boundaries , 1974 .