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


Snapshots


Details

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:

.

References

[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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