A goal-oriented hp-adaptive finite element method with electromagnetic applications. Part I: electrostatics

We describe the development and application of a finite element (FE) self-adaptive hp goal-oriented algorithm for elliptic problems. The algorithm delivers (without any user interaction) a sequence of optimal hp-grids. This sequence of grids minimizes the error of a prescribed quantity of interest with respect to the problem size. The refinement strategy is an extension of a fully automatic, energy-norm based, hp-adaptive algorithm. We illustrate the efficiency of the method with 2D numerical results. Among other problems, we apply the goal-oriented hp-adaptive strategy to simulate direct current (DC) resistivity logging instruments (including through casing resistivity tools) in a borehole environment and for the assessment of rock formation properties