ABSTRACT
For this approximation, we have the following theoretical a priori estimate for the errors eu := u — Uh , e\ \ — \h , and eq := q — qh (see Gunzburger and Hou [8], and also [2]):
||Veu||n + IK||rc + UVexlln < c{ ||V (u - ^ ) | | n + || V (A - 7T/i)||n + || 9 - X ^lrc },
for arbitrary approximations 'iph, G Hh and \h G Qh •
—A u — f in Q, u — 0 on d£l. (4.1)
The variational formulation of this problem seeks u G H = H q(Q) such that
(Vm, V ^)n = (/, <f>)n V0 G H. (4.2)
The discrete approximations Uh G Hh are determined by the Galerkin equations
(V « /i ,V ^ )n = (/, ^ ) n M<j>hEHh-(4.3)
The essential feature of this approximation scheme is the “Galerkin orthogonal ity” of the error e u — Uh ,
(Ve, V(f)h)n = 0, <j>h£Hh-(4.4)
Next, we seek to derive a posteriori error estimates. Let «/(•) be a linear “error functional” defined on H and z G H the solution of the corresponding dual problem
(V</>,Vz)n = V^G tf. (4.5)
Taking <j> — e in (4.5) and using the Galerkin orthogonality, cell-wise integration by parts (observing that z\q$i — 0 ) results in the error representation
J(e) = (Ve, V z ) n = (Ve, V (z - <f>h))n
where [Vii/J denotes the jum p of Vuh across the interelement boundary, and <j>h G Hh is an arbitrary approximation. In this relation the factors z — <j>h may be viewed as weights expressing the sensitivity of the error quantity J(e) with respect to changes of the equation residuals ( / + A Uh)\r and jum p residuals ^•[Vu/^idT . From the error identity (4.6), we infer the following a posteriori error estimate
with the cell residuals and weights
p P := h ^ W f + Auh\\T ,pQj, := ^h^3/2\\n-[Vuh]\\dT\du,
In view of the local approximation properties of finite elements, there holds
^ c / / i |m a x |V 2z| . (4-8)
We remark that in a finite difference discretization of the model problem (4.1) the corresponding “influence factors” behave like « m ax j \z\.
