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Excerpt from Comparison of Linear System Solvers Applied to Diffusion-Type Finite Element Equations
Ordinarily, the basis functions (bl, are chosen to be nonzero over just a few intervals of the mesh. The matrix A is then quite Sparse, having zero entries in positions i,j such that d), and are nonzero over disjoint intervals. For example, in one dimension, a typical choice for is the space of continuous piecewise linear functions satisfying the essential boundary conditions, and the basis functions (b1, are taken to be one at one node and zero at all others, as pictured below.
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