Optimal Lagrange and Hermite finite elements for Dirichlet problems in curved domains with straight‐edged triangles

Abstract

One of the reasons for the success of the finite element method in Solid Mechanics, among other Applied Sciences, is its versatility to deal with bodies of arbitrary shape. In case the problem at hand is modeled by second‐order partial differential equations with Dirichlet conditions prescribed on a curvilinear boundary, method's isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for polygonal or polyhedral domains and methods of order greater than one based on standard straight‐edged elements. However, besides algebraic and geometric inconveniences, the isoparametric technique is limited in scope, since its extension to degrees of freedom other than function values is not straightforward. In previous work the author introduced a simple alternative, which bypasses the above drawbacks without eroding qualitative approximation properties. Among other advantages, it can do without curved elements and is based only on polynomial Algebra. The purpose of this paper is two‐fold: On the one hand the mathematical foundations of this method are established, in the framework of a model Poisson problem solved with Lagrange elements; in particular optimal‐rate error estimates both in energy norm and in the mean‐square sense are demonstrated and a numerical comparison with the isoparametric technique is carried out. On the other hand this study is extended to Hermite elements with normal‐derivative degrees of freedom for solving the equation modeling the bending of clamped thin plates with a curvilinear edge.

Optimal Dirichlet‐condition enforcement on curved boundaries for Lagrange and Hermite FEM with straight‐edged simplexes

“…Besides other advantages of the method studied in this work pointed out in [2,3], more particularly over the isoparametric technique, the former is more accurate than the latter. In support to this assertion we next present some comparative numerical results.…”

“…Remark 1: If k ≥ 4 and Ω is not convex, qualitatively equivalent results can be proved if suitable numerical quadrature formulae are used. In [2] we proved error estimates in the L 2 -norm for our method in terms of O(h k+1 ) for any k ≥ 2 if Ω is convex and for k = 2 otherwise.…”

“…Here P is the nearest intersection with Γ of the straight line joining M and the vertex O T of T opposite to e T , as shown in Figure 1 for k = 3. In [2] we proved that such a construction of W h is feasible. Fig.…”

“…Our approach is particularly handy when combined with a Hermite finite element method having normal derivative DOFs to be prescribed on a curved boundary. Actually a method of this type was addressed in the presentation at GAMM 2019 (see also [2] and other references therein), besides Lagrange finite elements. For the sake of brevity here we illustrate our method in the case of conforming Lagrange FEs.…”

“…Section 18: Numerical methods of differential equations Noting that V h and W h have the same dimension, the following results were proven in [2]:…”