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.