Nested Newton Strategies for Energy-Corrected Finite Element Methods

Abstract

Energy-corrected finite element methods provide an attractive technique for dealing with elliptic problems in domains with re-entrant corners. Optimal convergence rates in weighted $L^2$-norms can be fully recovered by a local modification of the stiffness matrix at the re-entrant corner, and no pollution effect occurs. Although the existence of optimal correction factors is established, it remains open how to determine these factors in practice. First, we show that asymptotically a unique correction parameter exists and that it can be formally obtained as the limit of level dependent correction parameters which are defined as roots of an energy defect function. Second, we propose three nested Newton-type algorithms using only one Newton step per refinement level and show local or even global convergence to this asymptotic correction parameter.