An approach for combining the Ritz-Galerkin and finite element method
A combined method for solving the two dimensional elliptic model problem \[ -\partial /\partial x(\beta \partial u/\partial x)-\partial /\partial y(\beta \partial u/\partial y)=f,\quad(x,y)\in s;\quad u=g,\quad(x,y)\in \Gamma \] which has been widely used is studied. The combined method consists in dividing s by a circle \(\Gamma_ 0\) into 2 subdomains: a circular domain \(s_ 2\) contained in s and another domain \(s_ 1\) such that \(s=s_ 1+s_ 2\). Two quite different methods, the finite element method and the Ritz-Galerkin method, are used on \(s_ 1\) and \(s_ 2\), respectively. The use of combined methods is particularly important in problems with complicated boundaries or boundary conditions, or in problems with solutions which are not smooth enough or have singularities or in problems with unbounded solution domains. In such cases, a single method is often ineffective. This method has the advantages that it is valid for general nonhomogeneous elliptic boundary value problems and that it is reasonably simple to describe.
scientific article; zbMATH DE number 3860336