(1) (a) Discretize the 2D Poisson equation Vu(x, y) = Uxx + Uyy = = p(x,...

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(1) (a) Discretize the 2D Poisson equation Vu(x, y) = Uxx + Uyy = = p(x, y) with second-order accurate central differences with Ax = the given charge density. Ay=h. p(x, y) is (b) Express uij in terms of its four nearest neighbors and pij. (k+1) (c) Write down the Jacobi iteration formula for uj at iteration k + 1 in terms of the four nearest neighbor values UNN at iteration k, plus p. (k) (d) Define the residual = Tij h ((Vu)ij + Pij). Then write the Jacobi method in terms of the residual r(). Finally write down the (e) Gauss-Seidel and (f) SOR iteration formulas for (k+1) Uij . Use the notation to denote the residual with current values u of u (either updated or not) on the right-hand side. (1) (a) Discretize the 2D Poisson equation Vu(x, y) = Uxx + Uyy = = p(x, y) with second-order accurate central differences with Ax = the given charge density. Ay=h. p(x, y) is (b) Express uij in terms of its four nearest neighbors and pij. (k+1) (c) Write down the Jacobi iteration formula for uj at iteration k + 1 in terms of the four nearest neighbor values UNN at iteration k, plus p. (k) (d) Define the residual = Tij h ((Vu)ij + Pij). Then write the Jacobi method in terms of the residual r(). Finally write down the (e) Gauss-Seidel and (f) SOR iteration formulas for (k+1) Uij . Use the notation to denote the residual with current values u of u (either updated or not) on the right-hand side.