Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 8 / a / Solution

Use the full continuous piecewise linear space $V_h\subset H^1(0,1)$, including endpoint basis functions, because the boundary conditions are Neumann. DefineChoose $u_h^0$ as the $L^2$ projection of $u_0$ onto $V_h$. For each time step seek $u_h^{m+1}\in V_h$ withThis is the conforming Crank-Nicolson method. The reaction coefficient and the diffusion are both treated at the midpoint; integration by parts has removed the boundary flux through the homogeneous Neumann condition.