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A new mass lumping scheme for the mixed hybrid finite element method
Author(s) -
Younes Anis,
Ackerer Philippe,
Lehmann François
Publication year - 2006
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.1628
Subject(s) - mass matrix , discretization , finite element method , matrix (chemical analysis) , mathematics , tetrahedron , partial differential equation , mathematical analysis , numerical analysis , polygon mesh , mixed finite element method , geometry , physics , materials science , nuclear physics , neutrino , composite material , thermodynamics
Diffusion‐type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M ‐matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M ‐matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M ‐matrix for rectangular elements and can be an M ‐matrix for triangular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than π/2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors. Copyright © 2006 John Wiley & Sons, Ltd.