Analysis of a Filtered Time-Stepping Finite Element Method for Natural Convection Problems

Abstract

In this paper, we present, analyze, and test a novel low-complexity time-stepping finite element method for natural convection problems utilizing a time filter (TF). First, via a TF to postprocess the solutions of backward Euler (BE) schemes, we make a minimally intrusive modification to the existing codes to improve the time accuracy by one order. This also provides, at no extra complexity, an estimate of the temporal error,which is easy to construct a novel adaptive algorithm. Additionally, the TF can remove the overdamping of the BE scheme while remaining unconditionally energy stable. Hence, this paper addresses the question, how can one improve the time accuracy without increasing computational and cognitive complexity? Then long time stability and error estimates of BE plus time filter (BETF) with constant time stepsize are proved. Moreover, we construct adaptive algorithms by extending the approach to variable time stepsize, and we extend the methods to higher order algorithms. Finally, numerical tests confirm the convergence rates of our method and validate the theoretical results.