A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition. (June 2016)

• Record Type: Journal Article
• Authors: Gerritsma, Marc; Bochev, Pavel
• Is Part Of: Computers & mathematics with applications. Volume 71:issue 11(2016)
• Page Start: 2285
• Page End: 2300
• Publication Date: 2016-06
• Subjects: Least-squares -- Mimetic methods -- Spectral element method -- Mass conservation; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2016.01.033
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009[3] ), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009[4, Chapter 7] ). Here we address this problem by proving a new non-standard stability bound for the velocity–vorticity–pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximated by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. We also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.