. We want to solve the partial differential equation (PDE) \frac{\partial^{2} U(x, y)}{\partial x^{2}}+\frac{\partial^{2} U(x, y)}{\partial y^{2}}=3 x+2 y subject to Dirichlet boundary conditions with g1(x, y) = Y, 92(x,

y) = e®+y, and g3(x, y)figure below; h = k = 1/4).= x (see a. Discretise the partial differential equation using a centred difference scheme with error O(h2)in each dimension. Explain what a stencil is and present it for the given problem.[8] b. Use the stencil derived in (a) to present the system of equations to solve. Find the solutions for the problem at P1, P2 and P3.[9] c. Explain how to compute the numerical solution if the condition on g1 (left hand side of the triangle) changes to a Neumann boundary condition.[8]

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