Numerical assessment of stability of interface discontinuous finite element pressure spaces

• Published in: Computer methods in applied mechanics and engineering 2012-10, Vol.245-246, p.63-74
• Main Authors: Sousa, Fabricio S., Ausas, Roberto F., Buscaglia, Gustavo C.
• Format: Article
• Language: English
• Subjects: Algebra; Assessments; Constants; Discontinuous interpolants; Drops and bubbles; Eigenvalue problem; Eigenvalues; Exact sciences and technology; Finite element method; Finite elements; Fluid dynamics; Fundamental areas of phenomenology (including applications); Inf–sup condition; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematical models; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Nonhomogeneous flows; Numerical analysis. Scientific computation; Numerical stability; Physics; Sciences and techniques of general use; Shape functions; Stability
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2012.06.019

► Inf–sup constants of several formulations are evaluated by solving generalized eigenproblems. ► We prove the stability of two interface-discontinuous FE spaces designed for multiphase flows. ► We prove that the stabilized P1/P1 formulation does not become unstable if stabilization is removed in a band of elements. ► We prove that the mini-element formulation does not become unstable if the velocity bubbles are removed in a band of elements. The stability of two recently developed pressure spaces has been assessed numerically: The space proposed by Ausas et al. [R.F. Ausas, F.S. Sousa, G.C. Buscaglia, An improved finite element space for discontinuous pressures, Comput. Methods Appl. Mech. Engrg. 199 (2010) 1019–1031], which is capable of representing discontinuous pressures, and the space proposed by Coppola-Owen and Codina [A.H. Coppola-Owen, R. Codina, Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions, Int. J. Numer. Methods Fluids, 49 (2005) 1287–1304], which can represent discontinuities in pressure gradients. We assess the stability of these spaces by numerically computing the inf–sup constants of several meshes. The inf–sup constant results as the solution of a generalized eigenvalue problems. Both spaces are in this way confirmed to be stable in their original form. An application of the same numerical assessment tool to the stabilized equal-order P1/P1 formulation is then reported. An interesting finding is that the stabilization coefficient can be safely set to zero in an arbitrary band of elements without compromising the formulation’s stability. An analogous result is also reported for the mini-element P1+/P1 when the velocity bubbles are removed in an arbitrary band of elements.