Premium
Formulation and solution of hierarchical finite element equations
Author(s) -
Wiberg NilsErik,
Möller Peter
Publication year - 1988
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620260514
Subject(s) - finite element method , mathematics , convection–diffusion equation , galerkin method , discontinuous galerkin method , method of mean weighted residuals , mathematical optimization , eigenvalues and eigenvectors , penalty method , diffusion , mathematical analysis , physics , quantum mechanics , thermodynamics
The paper presents a general hierarchical formulation applicable to both elliptic and hyperbolic problems. Static and eigenvalue linear elastic problems as well as convection–diffusion problems are studied. The hierarchical formulation is well suited for adaptive procedures. For the convection‐diffusion problem the hierarchical approximation is made in time only. Different hierarchical functions are proposed for different types of problems. Both weighted residual and least‐squares formulations are applied. A combination of these two gives a penalty method with a constraint equation corresponding to the least‐squares method. A whole class of time integration formulae is obtained. These are all suitable for adaptive procedures owing to the hierarchical approximation in the time domain. If a linear discontinuous hierarchical base function is used in the Galerkin weak formulation, the method so obtained corresponds to the discontinuous Galerkin method in time and is especially suited for convection dominated problems. The streamline‐diffusion method is found to be the aforementioned penalty method. This paper also examines the sequence of nested equation systems that results from a hierarchical finite element formulation. Properties of these systems arising from static problems are investigated. The paper presents some new possibilities for iterative solution of hierarchic element equations, and different procedures are compared in a numerical example. Finally, a simple ID convection‐diffusion problem clearly shows that the proposed hierarchical formulation in time gives a stable and accurate solution even for convection dominated flow.