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A Mixed Finite Element Method for the Dirichlet Vector Laplacian in Three Dimensions

Ralf Hiptmair, Peiyang Yu, Tianwei Yu

Abstract

This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition. The Dirichlet condition disrupts the structure of the standard de Rham complex, requiring the vorticity to be sought in a non-standard function space to achieve well-posedness. We derive error estimates that confirm the numerically observed suboptimal convergence rates. In particular, by developing a discrete Caccioppoli-type inequality for discrete curl-harmonic functions, we prove $(k-1/2)$-th order convergence in the energy norm on general domains and $k$-th order convergence in $L^2$ on convex domains, where $k \ge 1$ denotes polynomial degrees of the finite element spaces. These results extend the previous two-dimensional analysis developed in [Arnold, D.N., Falk, R.S. and Gopalakrishnan, J., 2012. Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions. Mathematical Models and Methods in Applied Sciences, 22(9), p.1250024.]~to three-dimensional domains with general topology. As a direct application, a discretization of the Stokes problem in vorticity-velocity-pressure form is studied.

A Mixed Finite Element Method for the Dirichlet Vector Laplacian in Three Dimensions

Abstract

This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition. The Dirichlet condition disrupts the structure of the standard de Rham complex, requiring the vorticity to be sought in a non-standard function space to achieve well-posedness. We derive error estimates that confirm the numerically observed suboptimal convergence rates. In particular, by developing a discrete Caccioppoli-type inequality for discrete curl-harmonic functions, we prove -th order convergence in the energy norm on general domains and -th order convergence in on convex domains, where denotes polynomial degrees of the finite element spaces. These results extend the previous two-dimensional analysis developed in [Arnold, D.N., Falk, R.S. and Gopalakrishnan, J., 2012. Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions. Mathematical Models and Methods in Applied Sciences, 22(9), p.1250024.]~to three-dimensional domains with general topology. As a direct application, a discretization of the Stokes problem in vorticity-velocity-pressure form is studied.
Paper Structure (19 sections, 20 theorems, 128 equations, 4 figures, 1 table)

This paper contains 19 sections, 20 theorems, 128 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Well-posedness of the continuous formulation
  3. Preliminaries
  4. The L2 de Rham complex
  5. Some decompositions
  6. Norm equivalence on Sigma
  7. Well-posedness
  8. Discretization
  9. Preliminaries
  10. Regularity results
  11. Discrete de Rham complexes and interpolation approximations
  12. Some discrete decompositions
  13. A curl-elliptic projection
  14. Norm equivalence on Sigmah
  15. A priori error analysis
  16. ...and 4 more sections

Key Result

Lemma 2.1

Figures (4)

  • Figure 1: Sketch of the partition $\Omega = \bigcup_{j=0}^J A_j$
  • Figure 2: Case I: Convergence study of the scheme \ref{['eq:hodgelapdisc']} on a domain with void
  • Figure 3: Case II: Convergence study of the scheme \ref{['eq:hodgelapdisc']} on a convex domain
  • Figure 4: Case III: Convergence study of the scheme \ref{['eq:vvpdisc']} on a convex domain

Theorems & Definitions (44)

  • Lemma 2.1: scalar/vector potentials
  • proof
  • Corollary 2.2: Helmholtz decomposition
  • Theorem 2.3: Regular decomposition of ${\boldsymbol{H}_0(\mathrm{div})}$ hiptmair_2020
  • Lemma 2.4: norm equivalence on $\boldsymbol{\Sigma}$
  • proof
  • Remark 1: comparison with arnold_2012
  • Theorem 2.5: wellposedness
  • proof
  • Lemma 2.6: boundedness and inf-sup condition
  • ...and 34 more