Mass Lumping Emanating from Residual-free Bubbles

A nodally exact scheme is derived for a model equation in 1D involving zeroth and second order terms. The method is derived using residual-free bubbles in conjunction with the Galerkin approximation. It is shown that this approach leads to the mass lumping scheme for su ciently small mesh sizes. 1. The residual-free bubbles approach. The usage of the Galerkin method enriched with bubble functions has gained new impetus recently [1], [2]. In particular, the observation that this approach gives rise to streamline upwinding [2], [3], nally brought together two apparently distinct discretization procedures: namely, the employment of the standard Galerkin method with, possibly, more complex functions and the practice of combining the Galerkin method with least-squares like terms, viewed as upwind schemes. An abstract theory has been put together constructing a bridge between the stabilized methods and the Galerkin method using standard polynomial nite elements plus \virtual bubbles" [1]. Lately, a particular choice of bubble functions enabled to recover the one-dimensional nodally exact upwind scheme [3], [7], [8]. We will denote this choice by \residual-free" bubbles. The essential idea is that these bubbles are assumed to satisfy strongly the di erential equations in each element subjected to homogeneous boundary conditions on the element boundary. Let us consider a boundary value problem given by (1.1) ( Lu = f in u = 0 on = @ The work of the rst author was partially supported by the Istituto di Analisi Numerica del CNR, Pavia (Italy), and the work of the second author was partially supported by the ECC Project \The Equation of Fluid Dynamics and Related Topics" (HCM program). Typeset by AMS-TEX 1 where L is a second order linear elliptic di erential operator and f is a given function de ned on (we will omit technicalities, such as the limitations on the shape of , more complicated boundary conditions, etc., so not to encumber the presentation). Assume that problem (1.1) can be given a classical variational formulation as follows: (1.2) ( nd u 2 V such that (Lu; v) = a (u; v) = (f; v) for all v 2 V where a ( ; ) is a bilinear form on V = H 0 ( ) which is continuous and coercive with respect to the usual norm of V and ( ; ) is the usual scalar product in L( ). Let Vh V be a nite dimensional subspace of V ; then the Galerkin approximation for problem (1.2) is (1.3) ( nd uh 2 Vh such that a (uh; vh) = (f; vh) for all vh 2 Vh: The classical nite element method consists roughly in taking a partition Ph = fKg of and in de ning (1.4) Vh = V (k) h = fvh 2 C( ) : vhjK is a polynomial of degree kg: We wish to enrich the classical polynomial-based space V (k) h by adding a set of bubbles, i.e. functions whose support is contained within one element. These bubbles will be selected in such a way that the di erential equation is satis ed exactly in each element. For each K 2 Ph, let BK be a nite dimensional subspace of H 0 (K) (to be determined later); the functions of BK are to be thought as extended to zero outside K. De ne then (1.5) B = M K2Ph BK and (1.6) Vh = V (k) h B as the enlarged approximation space. Any function vh 2 Vh can then be split in a unique way as (1.7) vh = vk + vb = vk + X K2Ph vb;K