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

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.

Test with $v_h=u_h^{m+1}$ and apply $2(a-b,a)=\Vert a{\Vert }^2-\Vert b{\Vert }^2+\Vert a-b{\Vert }^2$. The exact energy identity isThe printed identity omits the last reaction contribution while explicitly describing its last term as the $H^1$ seminorm. It requires the full $H^1$ norm instead of that seminorm. With the printed seminorm its left side equals $-\Vert u_h^{m+1}{\Vert }_2^2$, rather than zero. The source reaction term forces this correction.

All terms other than the difference of successive squared norms are nonnegative. ConsequentlyThe second inequality uses contractivity of the initial orthogonal projection. No restriction relates $\Delta t$ to $h$, giving unconditional stability. The retained reaction term strengthens dissipation.

Define the mass matrix $M_{ij}={\int }_0^1{\varphi }_j{\varphi }_i$ and stiffness matrix $K_{ij}={\int }_0^1{\varphi }_j^{\prime }{\varphi }_i^{\prime }$ for $1\le i,j\le N$. With coefficient vectors $U^m$, the step system isThe matrices are symmetric, and $V^T\mathsf{A}V=\Vert v_h{\Vert }_2^2+\Delta t(\Vert v_h^{\prime }{\Vert }_2^2+\Vert v_h{\Vert }_2^2)>0$ for nonzero $V$, proving positive definiteness.

For $h=1/2$, the first basis function is the full tent at $1/2$, and the last is the half tent ending at $1$. Integration givesSince $\Delta t=2/2=1$, the requested matrix in this normalization isHere the matrix symbol $M$ is distinct from the printed integer number of time steps.