A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

Abstract

Abstract. We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated, and for many practical applications sharp explicit upper and lower bounds are readily obtained. The general results are then applied to finite element discretizations of scalar quasi-linear elliptic partial differential equations of 2nd order, the eigenvalue problem for scalar linear elliptic operators of 2nd order, and the stationary incompressible Navier-Stokes equations. They immediately yield a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.

A Posteriori Error Estimates for Non-Linear Problems

“…In [18] we developped a gênerai framework for the a posteriori error estimation of abstract noniinear problems. When applied to gênerai quasilinear elliptic équations of 2nd order it yields estimâtes on the W l ' r -norm of the error.…”

“…We then show that the residual || F( u h ) || y * is equivalent to || F k ( u h ) || ^ which, for the applications, will be much easier to compute. The main différence with [18] is the condition Y h c Y + . For applications, this means that the éléments of Y h must be of class C K with a suitable K 5* 1.…”

“…The main point is the construction of local eut-off functions, which are of class C K , and of a prolongation operator, which associâtes with a function of a face of a triangulation an extension that is defined on the whole U n and that is globally of class C K . The techniques of this section are inspired by those of [18]. But, due to the need for global C K -continuity, they considerably differ in details.…”

“…We obtain residual a posteriori error estimâtes which have the same structure as those given in [18] but which have different scaling factors. This is duc to the fact that in deriving the error estimâtes we irnplicitely use a duality argument.…”

“…We also dérive a posteriori error estimâtes which are based on the solution of auxiliary local Dirichlet problems. The auxiliary problems are the same as those used in [18]. But, now, the estimators are based on a Z/-norm of the solution of the auxiliary problem, instead of a W lljP -norm in [18].…”