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Sparse matrices can be useful for computing large-scale applications that dense matrices cannot handle. One such application involves solving partial differential equations by using the finite element method.

The finite element method is one method of solving partial differential equations (PDEs). Compared with another numeric method, the finite difference method, the finite element method can handle geometrically complicated domains straightforwardly. The finite element method also has the flexibility to deal with problems that vary rapidly. Using the finite element method to solve PDEs involves the following steps.

- Separate the problem domain into discrete elements.
- Formulate the PDE into an equivalent variational problem.
- Construct a finite dimensional subspace and determine its basis on the discrete elements.
- Use the basic functions in the variational problem and formulate it into a system of linear equations.
- Traverse all discrete elements and assemble the coefficient matrix and right-hand side.
- Solve the system of linear equations to get the solution of the original PDE.

The coefficient matrix is mostly sparse. Also, the size of the coefficient matrix is large in order to get an accurate approximation to the solution of PDEs. Therefore, practical finite element method applications always rely on sparse matrices and sparse matrix operations.

Complete the following steps to use the Multicore Analysis and Sparse Matrix VIs to solve the PDE by the finite element method.

- Compute the Delaunay triangulation of the problem domain using Delaunay Triangulation VI.
- Initialize a sparse matrix using Initialize Matrix VI. You can specify an appropriate maximum number of nonzeros based on problem size.
- Assemble the coefficient matrix using Set Matrix Subset VI.
- Reorder elements in the coefficient matrix according to their row and column indices using Reorder Elements VI. A sparse matrix with reordered elements can usually hit better performance in many sparse matrix operations.
- Solve the system of sparse linear equations using PARDISO Solver VI.

Refer to the Solve PDE by FEM VI in the labview\examples\Multicore Analysis and Sparse Matrix\Sparse Matrix\Solve PDE by FEM directory for an example that uses the LabVIEW Multicore Analysis and Sparse Matrix Toolkit to solve a Poisson equation by the finite element method.

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