Analysis of a class of two-scale finite element methods

Endre Suli
University of Oxford
Sunday 7th June 2015
Aula Seminari MOX-6° piano dip di matematica
The classical Galerkin finite element method exhibits spurious numerical oscillations when applied to a non-self-adjoint elliptic boundary value problem with dominant hyperbolic behaviour and the subscales in the problem (such as boundary layers) are not properly resolved. The aim of this talk is to discuss the analysis of a class of two-scale finite element methods based on the use of residual-free bubbles to capture the subscales. In the case of general finite element spaces consisting of continuous piecewise polynomials, the method is shown to exhibit an optimal rate of convergence. The analysis relies on a delicate function-space interpolation result due to Luc Tartar which states that (L_2(\Omega),H^1_0(\Omega)_{1/2,\infty}\supset (L_2(\Omega),H^1(\Omega))_{1/2,1}, with continuous embedding. The talk in based on joint work with Franco Brezzi and Donatella Marini (University of Pavia), and with my former D.Phil. student at Oxford, Andrea Cangiani (now also at the University of Pavia). References 1. F. Brezzi, D. Marini, and E. Suli. Residual-free bubbles for advection-diffusion problems: the general error analysis. Numerische Mathematik. 85 (2000) 1, 31-47. 2. A. Cangiani and E. Suli. Enhanced RFB method. Numerische Mathematik. Accepted for publication, 2005. 3. A. Cangiani and E. Suli. Enhanced residual-free-bubble finite element method. International Journal for Numerical Methods in Fluids. 47 (2005) 10-11, 1307-1313.