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Finite Element method

The mesh points of the grid are the vertices of triangles, located at points tex2html_wrap_inline1017 . We use a finite element discretization, introducing basis functions tex2html_wrap_inline1019 , which in the present work are piecewise linear over each triangle, and satisfy tex2html_wrap_inline1021 The basis function tex2html_wrap_inline1023 is equal to unity on vertex i, and equal to zero on all other vertices.

The variables in the MHD equations are represented as a sum over basis functions. We first consider discretization of the equations (7), (8), (9), and (10), in which the variables to be expanded in basis functions are the velocity stream function tex2html_wrap_inline987 the magnetic flux tex2html_wrap_inline991 vorticity tex2html_wrap_inline1031 and the current C. The variables in the MHD equations, such as tex2html_wrap_inline991 are represented as


We use a zero residual Galerkin approach in which the equations are multiplied by a basis function tex2html_wrap_inline1037 , and integrated over the domain. This gives the set of sparse matrix equations,




The matrices appearing in these equations are the mass matrix tex2html_wrap_inline1039 the stiffness matrix tex2html_wrap_inline1041 and the Poisson bracket tensor tex2html_wrap_inline1043 defined by


Both the stiffness and mass matrices are symmetric. The Poisson bracket is anti symmetric under the exchange of any two indices. This assures that some of the most important integral relations satisfied by the differential equations are preserved by the finite element discretization. This includes conservation of energy and magnetic flux in the absence of dissipation. The matrices are sparse, having nonzero elements only between those vertices connected by the side of a triangle.

The discretization of the current - vorticity forms of the MHD equations is similar and involves the same matrices.

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