First-Order System Least Squares on Curved Boundaries: Lowest-Order Raviart--Thomas Elements

Abstract

The effect of interpolated edges of curved boundaries on Raviart-Thomas finite element approximations is studied in this paper in the context of first-order system least squares methods. In particular, it is shown that an optimal order of convergence is achieved for lowestorder elements on a polygonal domain. This is illustrated numerically for an elliptic boundary value problem involving circular curves. The computational results also show that a polygonal approximation is not sufficient to achieve convergence of optimal order in the higher-order case. Introduction.Elliptic boundary value problems arising in science and engineering are often posed on domains with curved boundaries. For low-order finite elements it is usually sufficient to use a polygonal approximation of the boundary in order to ensure that the finite element order of convergence is not affected by the inaccurate representation of the boundary. For higher-order finite elements a more accurate representation of the boundary is required which is usually achieved by isoparametric finite elements; i.e., the same finite element space is used once more for the parametrization of a more accurate domain approximation. This is well-known in the case of standard nodal finite element spaces, and a complete theory can be found in many finite element books (see, e.g., [4,6]).Finite element methods which simultaneously approximate scalar (potential) and vector (flux) unknowns are also widely used due to the fact that improved approximation of fluxes is sometimes a desirable goal. Mixed and hybrid finite elements of saddle point structure [3] and first-order system least squares [2] are two popular approaches in this direction. Favorable conservation properties are among the advantages of the first of these classes of methods while the latter approach possesses an inherent error control and allows for simplified handling of coupling conditions. On the other hand, properties of either of these approaches are sometimes inherited by the other one due to closeness properties established in [5]. The treatment of curved boundaries in the context of edge-or face-based finite elements where Neumann boundary conditions are imposed on the normal flux is, however, rather seldom mentioned in the literature. A thorough analysis of the effect of polygonal boundary approximation on the Raviart-Thomas finite element approximations seems to be missing. The same appears to be true for the convergence analysis of parametric Raviart-Thomas elements in the higher-order case. General remarks in this direction can be found in the

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

“…The convergence analysis relies crucially on an estimate for the discrepancy of the normal flux associated with the parametric Raviart-Thomas spaces on the curved boundary if it is set to zero on its piecewise polynomial approximation. This estimate was already proved in [3] for the two-dimensional situation (and even earlier in [4] for the lowest-order case) and is extended here to three dimensions where some additional complications occur. Additionally, the treatment of inhomogeneous flux boundary conditions will be addressed based on a suitable interpolation operator for the parametric Raviart-Thomas space.…”

“…In the standard case, the two lines are almost on top of each other. For such a radially symmetric problem, the deviation from the optimal order of convergence is, however, also rather hard to observe, as we have noticed before in [4] for first-order system least squares. This is not so surprising if one keeps in mind that not only n • u but the entire u is set to zero on the boundary, therefore leaving much less room for effects of the inexactness of the boundary conditions.…”

“…which also finishes the proof of the lower bound (57). In order to illustrate the closeness of the functional and the actual error norm, we include a final example with a prescribed solution which we already considered in [4].…”

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].…”

An adaptive least‐squares mixed finite element method for the Signorini problem

“…A detailed analysis of the effect of this variational crime, using tools from , Sect. III.1] for the displacement and for the stress approximation would certainly be interesting but seems to be beyond the scope of this article. For the time being, our computational results in Section IV provide numerical evidence that the effect of approximating curved boundaries does also not effect the finite element convergence order.…”