A2.19 B2.20
(i) The five-point equations, which are obtained when the Poisson equation (with Dirichlet boundary conditions) is discretized in a square, are
where for all .
Formulate the Gauss-Seidel method for the above linear system and prove its convergence. In the proof you should carefully state any theorems you use. [You may use Part (ii) of this question.]
(ii) By arranging the two-dimensional arrays and into the column vectors and respectively, the linear system described in Part (i) takes the matrix form . Prove that, regardless of the ordering of the points on the grid, the matrix is symmetric and positive definite.
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