The maximum angle condition in the finite element method for monotone problems with applications in magnetostatics

Abstract

The finite element method for an elliptic equation with discontinuous coefficients (obtained for the magnetic potential from Maxwell's equations) is analyzed in the union of closed domains the boundaries of which form a system of three circles with the same centre. As the middle domain is very narrow the triangulations obeying the maximum angle condition are considered. In the case of piecewise linear trial functions the maximum rate of convergence $O(h)$ in the norm of the space $H^1(\Omega_h)$ is proved under the following conditions: 1. the exact solution $u\in H^1(\Omega)$ is piecewise of class $H^2$ ; 2. the family of subtriangulations $\{{\cal T}_h^{\rm A}\}$ of the narrow subdomain $\Omega^{\rm A}$ satisfies the maximum angle condition expressed by relation (38). The paper extends the results of [24].