The Finite Element Method with Nonuniform Mesh Sizes for Unbounded Domains

Abstract

The finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains. Consider the following model problem: \[ - \Delta u = f\quad {\text {in}}\;{\Omega ^C},\quad u = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial u}}{{\partial r}} + \frac {1}{r}u = o\left ( {\frac {1}{r}} \right )\quad {\text {as}}\;r = |x| \to \infty ,\] where ${\Omega ^C}$ is the complement in ${R^3}$ (three-dimensional Euclidean space) of a bounded set $\Omega$ with smooth boundary $\partial \Omega$, f and g are smooth functions, and f has bounded support. This problem is approximately solved by introducing an artificial boundary ${\Gamma _R}$ near infinity, e.g. a sphere of sufficiently large radius R. The intersection of this sphere with ${\Omega ^C}$ is denoted by $\Omega _R^C$ and the given problem is replaced by \[ - \Delta {u_R} = f\quad {\text {in}}\;\Omega _R^C,\quad {u_R} = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial {u_R}}}{{\partial r}} + \frac {1}{r}{u_R} = 0\quad {\text {on}}\;{\Gamma _R}.\] This problem is then solved approximately by the finite element method, resulting in an approximate solution $u_R^h$ for each $h > 0$. In order to obtain a reasonably small error for $u - u_R^h = (u - {u_R}) + ({u_R} - u_R^h)$, it is necessary to make R large. This necessitates the solution of a large number of linear equations, so that this method is often not very good when a uniform mesh size h is employed. It is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by $C{h^{ - 3}}$ with C independent of h and R.