2013/07/16 by Cameron Talischi, Gláucio H. Paulino, Talischi, Cameron +1
Engineering · Physics and Astronomy · #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1307.4423
openalex publication_date 2013/07/16 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Polygonal finite elements generally do not pass the patch test as a result of\nquadrature error in the evaluation of weak form integrals. In this work, we\nexamine the consequences of lack of polynomial consistency and show that it can\nlead to a deterioration of convergence of the finite element solutions. We\npropose a general remedy, inspired by techniques in the recent literature of\nmimetic finite differences, for restoring consistency and thereby ensuring the\nsatisfaction of the patch test and recovering optimal rates of convergence. The\nproposed approach, based on polynomial projections of the basis functions,\nallows for the use of moderate number of integration points and brings the\ncomputational cost of polygonal finite elements closer to that of the commonly\nused linear triangles and bilinear quadrilaterals. Numerical studies of a\ntwo-dimensional scalar diffusion problem accompany the theoretical\nconsiderations.\n