Realistic Eigenvalue Bounds for the Galerkin Mass Matrix

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Abstract
In time-dependent finite-element calculations, a mass matrix naturally arises. To avoid the solution of the corresponding algebraic equation system at each time step, ‘mass lumping’ is widely used, even though this pragmatic diagonalization of the mass matrix often reduces accuracy. We show how the unassembled form of finite-element equations can be used to establish (in an element-by-element manner) realistic upper and lower bounds on the eigenvalues of the fully consistent mass matrix when preconditioned by its diagonal entries. We use this technique to give specific results for a number of different types of finite elements in one, two, and three dimensions. The bounds are found by independent calculations on the elements, and, for certain element types, are independent of mesh irregularity. We give examples of when some of the bounds are attained. These results indicate that the preconditioned conjugate-gradient method is appropriate and very rapid for the solution of Galerkin mass-matrix equations.

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