A practical finite element approximation of a semi-definite Neumann ...

Numerische MathemaUk

Numer. Math. 51, 23-36 (1987)

9 Springer-Verlag 1987

A Practical Finite Element Approximation of a Semi-Definite N e u m a n n Problem on a Curved D o m a i n John W. Barrett 1 and Charles M. Elliott 2'*'** 1 Department of Mathematics, Imperial College, London SW7 z Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Summary. This paper considers the finite element approximation of the semi-definite N e u m a n n problem: - t 7. (a17u)=f in a curved domain f2 ~ R " #u ( n = 2 or 3), a ~ v = g on af2 and S u d x = q , a given constant, for d a t a f and f~

g satisfying the compatibility condition S f d x + ~ g d s = O . Q

Due to per-

0R

turbation of domain errors ( f 2 ~ f P ) the standard Galerkin approximation to the above problem may not have a solution. A remedy is to perturb the right hand side so that a discrete form of the compatibility condition holds. Using this approach we show that for a finite element space defined over D h, a union of elements, with approximation power h k in the L2 norm and with dist(f2, Qh)_m >_0 and k_> k' > 2

[w-~hwlm, < C h r - " l l w l l v , ~

VweHk'(z), V z 6 T h,

(2.6)

where C is a constant independent of h and w. The boundary 0 0 h is constructed with the following approximation properties. F o r each element zEB h there exists a local co-ordinate system (X~, Y~) such that X,eA~ and Y~eR, where A, is either an interval ( n = 2 ) or a triangle (n=3). The surface 0oh--012'C~r - is locally described by Y~=O~(X~). The surface dO is

A Practical Finite Element Approximation

27

locally described by Y+=qJ,(X~). We denote this section of 092 by Or?+. It is assumed that qJ+, ~'h~cl' 1(A+) and that they vanish at the vertices of A+. This immediately implies that

11_[7@hrllO, o~,A~Ch

II_F~h+N0,+,a+_ _ _ k - l ( 2 > k ) . However, this is not a sufficient condition, since (l.2b), (2.20) and (2.23) yield that .q

~h dx = - e - 1 Th(I). u d x - I us

(2.26)

~h

For a p p r o x i m a t i o n s satisfying (A1) and (A2), such as isoparametrics, it can be shown, see L e m m a 3.4, that in general IP(1)l < ChkEILflle, ~+ I1~112.~]

(2.27)

~h which and so the choice e = O ( h x) with 2 > k gives rise to an a p p r o x i m a t i o n u~ ~h m a y converge to u but does not even converge to u as h ~ 0 and with 2 < k u~ at a rate well below the optimal. The a p p r o x i m a t i o n of the penalised p r o b l e m (2.19) is advocated by Molchanov and G a l b a (1985) and their analysis shows that the optimal rate of convergence in the H l n o r m is achieved if e=O(h k-l) and any variational crimes present in the m e t h o d are such that IP(1)l= Clzl2 ~h

V z e S h.

(2.29)

(A4) For h < h o it is assumed that for all w~Hk((2) and for all w h and z e S h lah(wh, X)--ah( wh, x)l

< C {h k- 1 Ilwllk, r~h+ tlw --whll 1, ~h} II;tII1, ~h,

(2.30a)

--ah(wh, X)I