Finite-Element Approximation of Elliptic Equations with a Neumann or Robin Condition on a Curved Boundary

Abstract

This paper considers a finite-element approximation of a second-order selfadjoint elliptic equation in a region flcR" (with n = 2 or 3) having a curved boundary dQ on which a Neumann or Robin condition is prescribed. If the finite-element space denned over D h , a union of elements, has approximation power h k in the L 2 norm, and if the region of integration is approximated by Q* with dist (Q, £?*) =£ Ch k , then it is shown that one retains optimal rates of convergence for the error in the H 1 and L 2 norms, whether Q* is fitted (£^ = D h) or unfitted (Q* c £>*), provided that the numerical integration scheme has sufficient accuracy.

Finite element approximation of the Dirichlet problem using the boundary penalty method

“…Therefore the result (3.14) is a direct consequence of Lemma 5.10 in Barrett and Elliott [6]. Likewise the results (3.15) are a direct consequence of Lemmas 5.8 and 5.9 in Barrett and Elliott [6] and the trace inequality (1.17b).…”

“…Likewise the results (3.15) are a direct consequence of Lemmas 5.8 and 5.9 in Barrett and Elliott [6] and the trace inequality (1.17b). [] Remark 3.1.…”

“…Either partitioning O into elements if O is polyhedral or approximating O by O h, a union of isoparametric elements, if dO is curved. However, if QO is smooth and the imposed boundary condition is of the Neumann or Robin type; that is, (1.2b) is replaced by du a~-+ctu=g on dO, (1.6) dv where v denotes the outward pointing unit normal on dQ and ~(x)>0; then it is not necessary to fit the elements to the boundary in order to retain the optimal rate of convergence, see Barrett and Elliott [6]. They show that if the finite element space defined over D h, a union of elements, has approximation h K in the L 2 norm and if the region of integration is approximated by (P with dist(O, Oh)<Ch ~ then the optimal rate of convergence for the error in the H a and L 2 norms can be retained whether O h is fitted (Oh--D h) or unfitted (O h c Dh).…”

Isogeometric Analysis on V-reps: First results

“…The second term can be estimated, similarly as above, and with arguments similar to the ones used in [3,Lemma 4.2], as:…”

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

“…It is a simple matter to show, see Theorem 2.1 in Barrett and Elliott (1986), that the solutions of (1.4) and (1.20) satisfy in each element ~ ~ T* with _m(r ~ t3 t2)~ 0. The interpolation points being where ~?f2 crosses either the element's sides (n = 2) or edges (n = 3) yielding a polyhedral domain O h. A fuller description is given in Barrett and Elliott (1988). We now have…”