Past exam of the finite element methods for partial differential equations course of the University of Oxford / 2011 (C12.2b, DMAC 2B89, DMCO 2B89)

Original solutions to all three questions of the Finite Element Methods for Partial Differential Equations paper sat on 16 June 2011. Source: released Mathematics papers, Part C 2007 11.pdf. The changed integral in question 2 is interpreted on $(-1,1)$ with the original right-endpoint condition retained. Statements are not reproduced.

Use the usual bounded Lipschitz domain assumptions so that the Sobolev trace is defined. Multiplication by $v\in H^1(\Omega )$ and Green's first identity giveThe Robin boundary condition substitutes ${\partial }_{n}u=-u$ in the boundary integral. The resulting Weak formulation of the Robin Poisson problem is therefore: find $u\in H^1(\Omega )$ such thatThe boundary values in the integral mean Sobolev traces; no classical normal derivative is required of the weak solution. For uniqueness, subtract two solutions and test by their difference $w$. This yields ${\int }_{\Omega }|\nabla w{|}^2+{\int }_{\partial \Omega }{w}^2=0$. Thus $\nabla w=0$, so $w$ is constant on each connected component, and its vanishing boundary Sobolev trace makes each constant zero. The weak solution is unique.

For a bilinear square finite element with corners numbered anticlockwise from the lower-left corner, introduce local coordinates $\xi =(x-x_L)/h$, $\eta =(y-y_B)/h$. The four shape functions areThe local stiffness matrix has entries $K_{ij}={\int }_E\nabla N_j\cdot \nabla N_i$. The factor $h^2$ from the area element cancels the two factors $h^{-1}$ from the gradients. Symmetry leaves three types: diagonal, adjacent corners, and opposite corners. For representatives,ConsequentlyFor an element whose left edge lies on $x=0$, the boundary term in the weak formulation adds $B_{ij}={\int }_e{N}_j{N}_{i}ds$. Only $N_1=1-\eta$ and $N_4=\eta$ are nonzero on that edge. Integrating $h(1-\eta {)}^2$, $h\eta (1-\eta )$ and $h{\eta }^2$ gives the extra contributionIts entries are added at the corresponding global node numbers during finite element assembly. Additional boundary edges contribute their own analogous edge matrices. The Robin boundary condition is represented by these integrals, so boundary nodes remain unknowns.

A real bilinear form is a coercive bilinear form on $V$ when $a(v,v)\ge \alpha \Vert v{\Vert }_V^2$ for all $v\in V$ and some fixed $\alpha >0$. It is a bounded bilinear form, equivalently continuous, when $|a(v,w)|\le M\Vert v{\Vert }_V\Vert w{\Vert }_V$ for one finite $M$. A continuous linear functional satisfies $|\ell (v)|\le C_{\ell }\Vert v{\Vert }_V$.

The Galerkin method seeks $u_h\in S$ with $a(u_h,v_h)=\ell (v_h)$ for all $v_h\in S$. Subtraction from the continuous equations gives Galerkin orthogonality: $a(u-u_h,v_h)=0$ for every $v_h\in S$. Under the stated continuity and coercivity assumptions, Céa's lemma saysIndeed, with $e=u-u_h$ and any $v_h\in S$, Galerkin orthogonality eliminates $v_h-u_h$:If $e=0$ the claim is immediate; otherwise cancel $\Vert e{\Vert }_V$ and take the infimum. No symmetry of $a$ was used. These statements define an approximation when it exists; a finite-dimensional $S$ gives existence and uniqueness by the Lax-Milgram theorem, as does a closed vector subspace.

For the interval example write $L=c-b$ and $R=\max (|b|,|c|)$. The one-dimensional Sobolev representative of $v\in V$ satisfiesby the Cauchy-Schwarz inequality. The representation follows by taking the absolutely continuous function primitive of the weak derivative; its difference from $v$ has zero weak derivative, hence is constant, and $v(c)=0$ fixes that constant. Integrating the estimate gives the Endpoint Poincaré inequalityThus the nonnegative potential $x^2$ givesFor continuity, $|x|\le R$, the Cauchy-Schwarz inequality for the two integrals, and then its two-dimensional version giveSimilarly $\boxed{|\ell (v)|\le \Vert f{\Vert }_2\Vert v{\Vert }_2\le \Vert f{\Vert }_2\Vert v{\Vert }_V}$, proving continuity of the linear functional.

For the last displayed integral take $\Omega =(-1,1)$ and retain the endpoint condition $v(1)=0$, the domain-consistent interpretation of the changed formula. The changed form is still coercive. A negative potential on part of the interval does not by itself destroy coercivity. The sharper pointwise estimate above givesDiscarding only the positive part of the potential yieldsThe proved Endpoint Poincaré inequality with $L=2$ gives $\Vert v{\Vert }_V^2\le 3\Vert v^{\prime }{\Vert }_2^2$, and thereforeThis is a concrete application of negative-potential domination by endpoint control. The constant is sufficient, without claiming to be optimal.

Use a conforming finite element space $V_h\subset H_0^1(\Omega )$ with the prescribed basis, and write $u_h(x,y,t)={\sum }_j{U}_j(t){\varphi }_j(x,y)$. Testing the advection-diffusion equation by ${\varphi }_i$ and applying integration by parts to its diffusion term givesThusHere $M$ is the mass matrix; it is a symmetric positive-definite matrix because the basis is linearly independent. The initial vector represents the chosen finite element approximation to $g$, for example its orthogonal projection.

For any coefficient vector $U$, put $v_h={\sum }_j{U}_j{\varphi }_j$. The skew symmetry of Galerkin advection with constant drift $(1,1)$ givesbecause the Sobolev trace of $v_h$ vanishes. ConsequentlyAlong the solution, ${\int }_{\Omega }{u}_h^2=U^{T}MU$. The constant symmetric matrix $M$ and the semidiscrete equations therefore giveThe L2 norm of the spatial finite element solution is nonincreasing. The constant drift transports it without changing this energy; the diffusion term dissipates it. Any subsequent time-stepping method must be chosen to preserve an appropriate discrete stability estimate.