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Upwind basis finite elements for convection‐dominated problems
Author(s) -
Steffler P. M.
Publication year - 1989
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650090403
Subject(s) - upwind scheme , mathematics , finite element method , stream function , streamlines, streaklines, and pathlines , basis function , convection–diffusion equation , petrov–galerkin method , quadratic equation , vorticity , galerkin method , mathematical analysis , convection , flow (mathematics) , mechanics , geometry , physics , vortex , discretization , thermodynamics
Finite elements using higher‐order basis functions in the spirit of the QUICK method for convection‐dominated fluid flow and transport problems are introduced and demonstrated. Instead of introducing new internal degrees of freedom, completeness is achieved by including functions based on nodal values exterior and upwind to the element domain. Applied with linear test functions to the weak statements for convection‐dominated problems, a family of Petrov–Galerkin finite elements is developed. Quadratic and cubic versions are demonstrated for the one‐dimensional convection–diffusion test problem. Elements of up to seventh degree are used for local solution refinement. The behaviour of these elements for one‐dimensional linear and non‐linear advection is investigated. A two‐dimensional quadratic upwind element is demonstrated in a streamfunction–vorticity formulation of the Navier–Stokes equations for a driven cavity flow test problem. With some minor reservations, these elements are recommended for further study and application.