Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / B / 8 / i / Solution

Use $V=\{v\in H^1(0,1):v(0)=0\}$ and its continuous piecewise linear subspace $V_h=\mathrm{span}\{{\varphi }_1,\dots ,{\varphi }_N\}$. The condition at $x=1$ is natural, so the last hat function remains free. Set $u_h^0$ to the $L^2$ orthogonal projection of $u_0$ onto $V_h$: the stated $L^2$ initial data need not have nodal values.

For each $m$, find $u_h^{m+1}\in V_h$ such thatThis applies Backward Euler method to diffusion and reaction together. The step bilinear form is coercive, so each step has a unique solution.