The Weighted Vector Finite Element Method for Vector Wave Equation with Singularity

Boundary value problems for wave vector equations with singularity are used in mathematical models of electromagnetism problems in domains with a boundary containing reentrant corners. The solutions to such problems do not belong to the Sobolev space \(W^1_2\) and the approximate solution by the classical finite element method has a low rate of convergence to the exact solution. We define an \(R_\nu \) -generalized solution to the boundary value problem for a wave vector equation with corner singularity in a set of the weighted Sobolev-Monk space. A weighted vector finite element method (WVFEM) is constructed to find an approximate \(R_\nu \) -generalized problem. The WVFEM basis functions contain weighting functions to a degree depending on the sizes of the reentrant corners at the domain boundary. This allows us to weaken the influence of the singularity on the accuracy of finding the solution. The method has a convergence rate of O(h) in the norm of the weighted space \(L_{2,\alpha }(\varOmega )^2\) . Numerical calculations of model problems confirmed the theoretical estimate of the convergence rate.