Mass Matrix Computation in the Finite Element Method

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This Demonstration shows the computation of the mass matrix in a particular example of the finite element method. It considers piecewise linear basis functions. You can explore all the cross products of basis functions elementwise in a very simple mesh.

Contributed by: Mikel Landajuela (July 2018)
Open content licensed under CC BY-NC-SA



Consider the rectangular domain and the finite-element mesh composed of five nodes and three elements as shown in the top-left part of the results.

The five piecewise linear basis functions associated to each of the nodes are shown in the results.

The mass matrix is defined as


The finite element approximation reads: Find such that , where is the stiffness matrix () and is the mass matrix.

In finite-element programming, the computation of this matrix is usually performed elementwise, looping over all the elements and adding the nonzero contributions to the global matrix:

Many of the matrix elements are zero, so that reduces to:



[1] M. G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer-Verlag: Berlin, Heidelberg, 2013.

[2] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag: New York, 2004.

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