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Task 1 Consider the following problem Uxx + uyy =0, (x,y) ES = (0,1) X (0,1) u(x,0) = x, u(x,1) =1-x,u(0,y)= y, = = u(1,y) = 1 - y, (x,y) € as The region Qh is subdivided into a uniform grid of size h = 1/N. The following finite difference scheme is proposed to solve the elliptic equation on the unit square. = - + hz/ue,m+1 - 2ve,m + ve,m-1] = 0 1.2 (by hand) Motivate if the Jacobi iteration method applied to the linear system in 1.1 will converge to a unique solution. Task 2 Let S2 be the unit square (0,1) X (0,1), with boundary 89. The elliptic problem a2u a2u Lu ax2 8y² = with boundary conditions u(x,y) = g(x,y) for (x,y) € as is approximated in the interior S2 by a uniform grid points at distance h = 1/N. The intersections are (xe,3m) with x = lh and ym = mh for e,m = 0,1,2, N. Consider the scheme L^ve,m = 13/18/1,m - = f(xe,ym) 2.1 Prove that L've,m < 0 Oh anch 2.2 Given that this method satisfies the maximum principle ve,m > 0 for all = max ve.m. < max ve.m. sh anh verify that the solutions ve,m is stable and that the error |u(xe,ym) - ue,ml is of order h2.

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