Adaptive hp-Finite Element Method for Electromagnetic Field Logging Problems

A novel, highly efficient and accurate adaptive higher-order finite element method (hp-FEM) is proposed for electromagnetic field problems. Presented in this paper are the vector expression of Maxwell's equations, three kinds of boundary conditions, stability weak formulation of Maxwell's equations, and automatic hp-adaptivity strategy. This method can select optimal refinement and calculation strategies based on the practical formation model and error estimation. Numerical experiments show that the new hp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom, which provides more accurate results than those obtained using the adaptive h-FEM. The methodology is freely available online in the form of a general public licensed C++ library Hermes (http://hpfem.org/hermes).