Determination of the modulus of quadrilaterals by finite element methods (Q919477)

The aim of the paper is to establish a method of finite element approximations by which the modulus of quadrilaterals on Riemann surfaces can be determined. The boundary of the quadrilateral Q is supposed to be compact and piecewise analytic. The characteristic of the method is the use of ordinary triangular meshes and linear elements on parametric disks that are mapped on subregions of Q. The corresponding approximating functions satisfy the boundary conditions exactly even in the case of curvilinear boundary arcs, and express singular property exactly near inner and corner singularities. This results in optimal error bounds (e.g. \(O(h^ 2)\) for the modulus of Q). Since these comparable approximations are hard to compute a class of modified approximating functions is introduced. It is shown that this modification does not change the order of convergence. Numerical calculations of the modulus of some concrete quadrilaterals show fairly good results also for the case of plane domains.