Optimal Recovery in the Finite-Element Method, Part 1: Recovery from Weighted L2 Fits

Finite-element methods lead to approximations that are optimalor near-optimal in an associated energy norm. The consequentialproblem of recovering point values of the solution or its derivativesis addressed in this paper and its companion. A general frameworkis adopted, based on the seminal paper of Golomb & Weinberger(1959), which brings together several ideas such as superconvergence,local averaging and defect correction. Detailed results aregiven for piecewise constant and 2 linear approximations inone dimension: recovery from weighted L² fits is treated here,and from weighted H¹ fits in Part II     

Optimal Petrov--Galerkin Methods through Approximate Symmetrization

^*Present address: Department of Mathematics, Imperial College, London SW7 2BZ. A technique of approximate symmetrization is used to derivea test space from a given trial space for a Petrov—Galerkinmethod. This is applied to one-dimensional diffusion—convectionproblems to give approximations which are near optimal in anenergy norm. Rigorous and precise error bounds are derived todemonstrate the uniformly good behaviour and near optimalityof the procedure over all values of the mesh Péclet number.     

A Finite Element Moving Boundary Computation with an Adaptive Mesh

The finite element method is used to obtain an accurate approximationto the solution of a well-known moving boundary problem. Thealgorithm uses the projective nature of the approximate solutionto follow the movement of the boundary, and the element notesare adapted as the region of the solution changes.     

The Use of Divided Differences in Finite Element Calculations

Divided differences are used to calculate gradients of functionscomputed by finite element methods. On regular elements it isshown that for quite general difference formulae and commonfinite element methods this retains the full order of accuracyof the methods. Examples are given together with the resultsof numerical trials. Applicability to irregular meshes is consideredand a practical test with encouraging results is given.     

Discretization of a convection-diffusion equation

The unsteady convection-diffusion equation with constant coefficientsadmits an exact solution in the form of a convolution integral,which provides an explicit representation of the evolution operatorthrough one time step. This is used to unify many numericalschemes for the equation, showing interrelationships betweenfinite difference and finite element schemes and presentinga general framework for detailed error analysis. In particular,the upwind scheme, Lax Wendroff, QUICKEST, the ECG schemes andCrank Nicolson are all members of a family that includes powerfulnew schemes. Fourier analysis is used to obtain practical stabilityregions and some insights into accuracy; and the Peano kerneltheorem is also used to derive rigorous error bounds that canbe generalized to irregular meshes.     

Finite Volume Methods and their Analysis

Finite volume methods are widely used and highly successfulin computing solutions to conservation laws, such as those occurringin fluid dynamics: but little analysis of their behaviour hasbeen carried out. In this paper we use model problems in oneand two dimensions to initiate a study of such methods, especiallythe cell vertex method. In one dimension it shows that finitevolume methods give accurate flux values at volume boundaries:thus, even for the self-adjoint equation (a?'=f, these gradientvalues can be more accurate than for finite element methods.The cell vertexes schemes are aimed at convection dominatedproblems and have highly advantageous properties for convection-diffusionproblems: but we give error bounds for these four-point approximationsto the pure diffusion problem in one dimension which indicatestheir remarkable robustness. Finally, an analysis of the cellvertex scheme for the pure advection problem in two dimensionsis given. It explains the insensitivity of the method to meshstretching in the coordinate directions, maintaining second-orderaccuracy on any mesh.     

Optimal Recovery in the Finite-Element Method, Part 2: Defect Correction for Ordinary Differential Equations

This paper continues the study of optimal recovery formulaefor piecewise linear approximations to an unknown function u(x):here recovery is with respect to the mixed norm which occurs naturally when u arises from a Sturm-Liouvilleproblem. It is shown that local recovery is possible only whenp = O(qh²) Otherwise, global recovery is required and this isachieved by a defect-correction approach, closely related tothe difference-correction techniques of Fox used with finite-differencemethods. Results on the ensuing improvement of accuracy aregiven which show that typically O(h⁴) can be achieved with onesimple iteration.