Parametric Raviart--Thomas Elements for Mixed Methods on Domains with Curved Surfaces

Abstract

The finite element approximation on curved boundaries using parametric Raviart-Thomas spaces is studied in the context of the mixed formulation of Poisson's equation as a saddlepoint system. It is shown that optimal order convergence is retained on domains with piecewise C k+2 boundary for the parametric Raviart-Thomas space of degree k ≥ 0 under the usual regularity assumptions. This extends the analysis in [3] from the first-order system least squares formulation to mixed approaches of saddle-point type. In addition, a detailed proof of the crucial estimate in three dimensions is given which handles some complications not present in the two-dimensional case. Moreover, the appropriate treatment of inhomogeneous flux boundary conditions is discussed. The results are confirmed by computational results which also demonstrate that optimal order convergence is not achieved, in general, if standard Raviart-Thomas elements are used instead of the parametric spaces.

Least-Squares Methods for Linear Elasticity: Refined Error Estimates

“…A simple polygonal design of an exact displacement with homogeneous boundary conditions on ∂Ω implies For the linear elasticity problems, we compared the approximations obtained by the Least-Squares finite element method with the approximations obtained by the standard conforming finite element method and the mixed finite element method and prove that the H 1 -conforming displacement approximations (least-squares finite element and standard finite element) as well as the H(div)-conforming stress approximations are higher-order perturbations of each other. Future work will consider domain with curved boundaries in the spirit of [5,4,6,1].…”

Weakly symmetric stress equilibration for hyperelastic material models

“…The inf‐sup is therefore equivalent to the existence of a constant β > 0 such that with the mapped Nédélec space Ξ h , z φ holds. This is exactly the inf‐sup condition for the original spaces from Boffi et al in mapped coordinates using parametric Raviart‐Thomas elements for the stress approximation.…”

“…ℎ, denotes the 2 ( )-orthogonal projection to 0 ℎ, , then (28) implies that ℎ, =  ,0 ℎ, ( 1 1 + 2 2 + 3 3 ) + which means that ℎ, =  ℎ, ( 1 1 + 2 2 + 3 3 + ̃ ) with some ̃ ∈ I R 3 . Since all rigid body modes ∈ ( ℎ ) which can be written as = 1 1 + 2 2 + 3 3 + ̃ , we have the corresponding representation of ℎ, in (23).…”

Analysis of Mixed Finite Elements for Elasticity. I. Exact stress symmetry

“…Here we consider an element K ∈ C h with the vertices V 0 , V 1 and V 2 (the triangle filled with gray color). Now let Ψ K ∈ P k+2 (K) with Ψ K (K) = K be a polynomial mapping from K to the curved triangle K (filled with orange color), where we have chosen the order k + 2 as suggested in [9]. Then, in order to guarantee normal continuity, see (3.2), the stress finite elements are mapped by a Piola transformation, see [11], which includes the mapping Ψ K .…”

“…where D(•) denotes the Jacobian. For more details we refer to [8,9]. Note that the mapping Ψ K • Φ K is applied for all sub triangles as illustrated in Figure 1.…”