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Triangle Based Matrices

The matrices appearing in (23), (24), and (25) can be calculated analytically in each triangle and accumulated on the triangle vertices. The matrices have a simple form locally in each triangle. In each triangle, introduce a local numbering of the vertices tex2html_wrap_inline1045 tex2html_wrap_inline1047 labeled in a counter clockwise order. The side segments tex2html_wrap_inline1049 are labeled by their opposite vertex,


and so on in cyclic order, or


where the permutation symbol tex2html_wrap_inline1053 if the values of the indices are in ascending order or a cyclic permutation, = -1, if the indices are in descending order or a cyclic permutation, or = 0, if any two indices have the same value.

The triangle area tex2html_wrap_inline1059 is given by


The basis functions can be expressed as


To calculate the matrices, we need integrals of basis functions over the triangle, which are


The contribution to the mass matrix from a single triangle, tex2html_wrap_inline1063 is


To calculate the stiffness matrix and Poisson bracket tensor, we need the gradients of the basis functions, which are constant in each triangle, and from (28) are given by


This immediately yields the the contribution to the stiffness matrix from a single triangle, tex2html_wrap_inline1065


The contribution in a triangle to the Poisson bracket tensor is given by tex2html_wrap_inline1067


which is independent of tex2html_wrap_inline1069

The local matrices are assembled globally by summing the contribution of each triangle which shares a given vertex i.

It is convenient to consider a diagonalized form of the mass matrix. The lumped mass matrix is formed by subtracting all the off diagonal matrix elements in each row and adding them to the diagonal,


where a(i) is the label of a triangle having vertex i. Its value, from (29), is one third the area of the triangles surrounding the vertex.

Using the lumped mass, the finite element discretization is equivalent to a finite volume discretization, where the control volumes are constructed by joining the barycenters of the triangles (average of the vertex positions) to the midpoints of the triangle edges.

next up previous
Next: Laplacian Up: An Adaptive Finite Previous: Finite Element method

Hank Strauss
Wed Jan 7 14:07:46 EST 1998