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A variant of the Raviart-Thomas method for smooth domains with straight-edged triangles

Fleurianne Bertrand, Vitoriano Ruas

TL;DR

This work develops a Petrov-Galerkin variant of the Raviart-Thomas RT$_k$ mixed finite-element method on straight-edged triangles for smooth 2D domains with Neumann boundary parts. By using asymmetric test/shape-flux spaces, the authors prove uniform $H(div)$-type stability via a trio of inf-sup conditions and derive interpolation and a priori error estimates, yielding uniform convergence and, in many cases, optimal rates. They show how to handle curved-domain boundary effects without curved elements and discuss extensions to nonconvex domains, mixed boundary conditions, and potential 3D generalizations, including BDM compatibility. The paper also proposes a boundary-curving adjustment to recover optimal orders for arbitrary $k$ on mixed BCs, supported by theoretical bounds and numerical evidence. Overall, the approach provides a robust, higher-order, geometry-accurate framework for mixed finite elements on curved boundaries with practical relevance for elliptic PDEs.

Abstract

Several physical problems modeled by second-order elliptic equations can be efficiently solved using mixed finite elements of the Raviart-Thomas family RTk for N-simplexes, introduced in the seventies. In case Neumann conditions are prescribed on a curvilinear boundary, the normal component of the flux variable should preferably not take up values at nodes shifted to the boundary of the approximating polytope in the corresponding normal direction. This is because the method's accuracy downgrades, which was shown in previous papers by the first author et al. In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this article an alternative with straight-edged triangles for two-dimensional problems is proposed. The key point of this method is a Petrov-Galerkin formulation of the mixed problem, in which the test-flux space is a little different from the shape-flux space. After describing the underlying variant of RTk we show that it gives rise to a uniformly stable and optimally convergent method in the natural norm, taking the Poisson equation as a model problem.

A variant of the Raviart-Thomas method for smooth domains with straight-edged triangles

TL;DR

This work develops a Petrov-Galerkin variant of the Raviart-Thomas RT mixed finite-element method on straight-edged triangles for smooth 2D domains with Neumann boundary parts. By using asymmetric test/shape-flux spaces, the authors prove uniform -type stability via a trio of inf-sup conditions and derive interpolation and a priori error estimates, yielding uniform convergence and, in many cases, optimal rates. They show how to handle curved-domain boundary effects without curved elements and discuss extensions to nonconvex domains, mixed boundary conditions, and potential 3D generalizations, including BDM compatibility. The paper also proposes a boundary-curving adjustment to recover optimal orders for arbitrary on mixed BCs, supported by theoretical bounds and numerical evidence. Overall, the approach provides a robust, higher-order, geometry-accurate framework for mixed finite elements on curved boundaries with practical relevance for elliptic PDEs.

Abstract

Several physical problems modeled by second-order elliptic equations can be efficiently solved using mixed finite elements of the Raviart-Thomas family RTk for N-simplexes, introduced in the seventies. In case Neumann conditions are prescribed on a curvilinear boundary, the normal component of the flux variable should preferably not take up values at nodes shifted to the boundary of the approximating polytope in the corresponding normal direction. This is because the method's accuracy downgrades, which was shown in previous papers by the first author et al. In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this article an alternative with straight-edged triangles for two-dimensional problems is proposed. The key point of this method is a Petrov-Galerkin formulation of the mixed problem, in which the test-flux space is a little different from the shape-flux space. After describing the underlying variant of RTk we show that it gives rise to a uniformly stable and optimally convergent method in the natural norm, taking the Poisson equation as a model problem.
Paper Structure (21 sections, 29 theorems, 260 equations, 2 figures, 1 table)

This paper contains 21 sections, 29 theorems, 260 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The model problem and its abstract Petrov-Galerkin approximation
  3. Meshes and related sets, notations and assumptions
  4. The finite-element approximation method
  5. Well-posedness and uniform stability of the approximate problem
  6. Uniform inf-sup condition for bilinear form $a_h$ on $Ker(div)$
  7. Uniform inf-sup condition for bilinear form $b_h$
  8. Uniform inf-sup conditions for bilinear forms $d_h$ and $c_h$
  9. Interpolation errors
  10. Error estimates
  11. The convex case
  12. The non convex case
  13. Recovering optimal orders for mixed boundary conditions with arbitrary $k$
  14. Final remarks and conclusions
  15. The case of pure Dirichlet conditions
  16. ...and 6 more sections

Key Result

Proposition 3.1

If Assumption$^{*}$ holds, there exists a constant $C_{0}$ depending only on $\Gamma$ such that $\forall M \in e_T$ the length of the segment perpendicular to $e_T$ joining $M$ and $N \in \Gamma_T$ is bounded above by $C_{0} h_T^2$.

Figures (2)

  • Figure 1: Normal-flux DOFs shifted from $\Gamma_{1,h}$ to $\Gamma_1$ for $RT_1(T)$ (left) and $RT_0(T)$ (right) with $T \in {\mathcal{S}}_{1,h}$
  • Figure 2: A portion of $\Omega$ and $\Omega_h$ and the attached polar coordinate systems with origin $O \in \Omega$.

Theorems & Definitions (29)

  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • Proposition 4.2
  • Lemma 4.3
  • Theorem 5.1
  • Proposition 5.2
  • Theorem 5.3
  • Proposition 5.4
  • Proposition 5.5
  • ...and 19 more