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On a uniform $d$-dimensional spatial grid, the discrete L2 norm weights the squared Euclidean norm by the cell volume. It approximates the continuous L2 norm and is induced by the discrete L2 inner product. A contraction bound in this norm with a mesh-independent constant proves stability of a numerical method for the spatially discretized evolution.

The global error is the difference between the computed and exact solutions after accumulated numerical steps, rather than the local truncation error of a single step started from exact data. An estimate for the stability of a numerical method controls how initial errors and local defects accumulate. For a stable order-$p$ time integrator with starting errors $O(h^p)$, local defects $O(h^{p+1})$ accumulated over $O(1/h)$ steps give global error $O(h^p)$ on a fixed time interval.

The original PDF puts both neighbouring values at time $n+1$. Thus the actual scheme is Backward Euler method in time with a centered three-point approximation to the second spatial derivative. The old-time neighbours in the TeX transcription describe a different method.

Let $h=\Delta x$ and $k=\Delta t$. Insert a sufficiently smooth exact solution and divide by $k$. Expanding at $(x_m,t_{n+1})$ givesThe PDE cancels the $u_t-u_{xx}$ term. Therefore the normalized local truncation error is $O(k+h^2)$ andFor compatible smooth data with exact nodal initialization, the stability of a numerical method established below gives global error $O(k+h^2)$ on fixed time intervals, by summing the one-step defects with the contraction bound. Thus no relation between $k$ and $h$ is needed for stability; choosing $k=O(h^2)$ makes both consistency contributions $O(h^2)$.

Let $T$ be the $M\times M$ tridiagonal matrix with diagonal two and adjacent entries minus one, incorporating zero Dirichlet boundary conditions. The actual PDF scheme isAn orthonormal basis of eigenvectors of $T$ has components $\sqrt{2/(M+1)}\sin (jm\pi /(M+1))$, and substitution givesThey form an orthonormal basis by the spectral theorem for real symmetric matrices. Hence $I+\mu T$ is invertible for every $\mu >0$, and the amplification eigenvalues are $g_j=(1+\mu {\lambda }_j{)}^{-1}\in (0,1)$. For the discrete L2 norm $\Vert U{\Vert }_h^2=h{\sum }_m|U_m{|}^2$,This bound has constant one independent of time step and mesh, which is the required stability of a numerical method. Therefore every positive Courant number is stable. If a forcing or local-defect sequence is added, the discrete variation-of-constants formula gives a bound by the initial norm plus the sum of those perturbation norms; this also justifies the global error conclusion in part (a).

Arrange the interior values into a vector $U$. Represent the five-point Dirichlet Laplacian as a Kronecker sum $L_h=h^{-2}(D\otimes I+I\otimes D)$, where $D$ is the tridiagonal matrix with diagonal $-2$ and adjacent entries one. This Kronecker sum is real symmetric; the real sampled potential has a real diagonal matrix $D_V$. ThusThe generator $-i{H}_h$ is a skew-Hermitian matrix. ConsequentlyThe matrix exponential is a unitary matrix, either by differentiating its product with its adjoint or by unitary diagonalization of a normal matrix. For the two-dimensional discrete L2 norm $\Vert U{\Vert }_h^2=h^2{\sum }_{m,n}|U_{m,n}{|}^2$, this givesApplying the same equation to a difference proves stability of a numerical method with a mesh-independent constant one. This is norm conservation of a semidiscrete Schrödinger equation. It concerns continuous time after spatial discretization; an arbitrary subsequent time integrator need not preserve this stability.

The printed coordinates do not discretize the stated square. For $[-1,1{]}^2$ with $M$ interior points in each direction, use $h=2/(M+1)$, $x_m=-1+mh$, $y_n=-1+nh$, and sample $V$ there. The printed $h=1/(M+1)$ and unshifted coordinates instead describe a grid on $[0,1{]}^2$. The matrix proof is valid for either geometry with its corresponding boundary values, so this transcription-independent statement flaw does not alter the stability conclusion.

The starting point of a conforming finite element method is a weak formulation in a Hilbert space $X$. For example, a homogeneous Dirichlet boundary condition on a bounded Lipschitz domain gives $X=H_0^1(\Omega )$. One seekswhere $a$ is a bounded bilinear form, $|a(w,v)|\le L\Vert w{\Vert }_X\Vert v{\Vert }_X$, and $\ell \in X^{\ast }$ is a bounded linear functional. A conforming finite element space $X_h\subset X$ is finite dimensional and is typically built from functions that are polynomial on each element of a finite element mesh. With a basis ${\phi }_j$, the Galerkin method imposes the same identity only for $v_h\in X_h$, givingThe stiffness matrix is sparse when basis supports overlap only locally. Local element integrals assemble its entries; the mass matrix represents an $L^2$ term. This separates the choice of trial functions from the variational principle determining their coefficients.

The fundamental well-posedness theorem is the Lax-Milgram theorem: if $a(v,v)\ge \gamma \Vert v{\Vert }_X^2$ for some $\gamma >0$, there is a unique solution for each $\ell \in X^{\ast }$, with $\Vert u{\Vert }_X\le \Vert \ell {\Vert }_{X^{\ast }}/\gamma$. The same theorem on $X_h$ gives a unique discrete solution. The constants must be independent of the mesh for uniform stability of a numerical method. Boundary conditions are built into the trial space when they are essential constraints; flux conditions arise through boundary terms in integration by parts and are natural constraints in the weak formulation.

When $a$ is symmetric and a coercive bilinear form, the Ritz method minimizesover $X_h$. Its first variation is exactly the Galerkin method, so Ritz minimization and Galerkin testing coincide for symmetric coercive problems. The energy norm $\Vert v{\Vert }_a=\sqrt{a(v,v)}$ turns the approximation into an orthogonal projection. Indeed, subtracting the exact and discrete weak equations gives Galerkin orthogonality, $a(u-u_h,v_h)=0$. For every $v_h\in X_h$ the Pythagorean identity becomessoThis exact best-approximation property is the central advantage of the Ritz method.

Symmetry is unnecessary for the Galerkin method. For any bounded, coercive form, the Céa lemma givesTo prove it, put $e=u-u_h$. Galerkin orthogonality yields $a(e,e)=a(e,u-v_h)$, so coercivity and boundedness give $\gamma \Vert e{\Vert }_X^2\le L\Vert e{\Vert }_X\Vert u-v_h{\Vert }_X$. Divide when $e\ne 0$ and take the infimum. Thus approximation capability plus uniform coercivity gives convergence. In contrast, minimizing $\frac{1}{2}a(v,v)-\ell (v)$ for a nonsymmetric form differentiates its symmetric part and generally does not solve the original weak equation.

A concrete symmetric example is the one-dimensional reaction-diffusion problem from Question 5. Continuous piecewise-linear hat functions give the diffusion stiffness matrix $S$ with diagonal $2/h$ and neighbouring entries $-1/h$, and the mass matrix $M$ with diagonal $2h/3$ and neighbouring entries $h/6$. Therefore $K=S+M$ and $F_i=i{h}^2$. Its positive-definite matrix is the coordinate form of the continuous energy. The exact smooth solution $u(x)=x-\sinh (x)/\sinh 1$ makes this an explicit test of the method, not just an abstract existence result.

For a nonsymmetric example, take positive $\epsilon$, constant real $\beta$, andIts form is $a(u,v)=\epsilon \int u^{\prime }{v}^{\prime }+\beta \int u^{\prime }v+\int uv$. Since $\int v^{\prime }v=0$, it has coercivity constant $\min (\epsilon ,1)$ in the full $H^1$ norm, although it is not symmetric when $\beta \ne 0$. The Galerkin method remains well posed. On the uniform hat basis, $K=\epsilon S+\beta C+M$, where $C_{i,i+1}=1/2$, $C_{i,i-1}=-1/2$ and $C_{ii}=0$. The skew-symmetric matrix $C$ cancels from $c^{T}Kc$, proving nonsingularity. A Ritz energy would omit precisely this advection contribution. Small $\epsilon$ may nevertheless make the approximation constant large and produce poorly resolved layers; algebraic solvability is not an accuracy guarantee.

Quantitative convergence uses a finite element interpolation estimate. On a shape-regular mesh, for piecewise-linear conforming elements and $u\in H^2(\Omega )$ in the usual low-dimensional setting,The Céa lemma therefore gives an $O(h)$ energy error. The Aubin–Nitsche duality argument improves the L2 norm error when the dual elliptic problem has $H^2$ regularity: solve $a(v,z)=(e,v{)}_{L^2}$ and useConsequentlywith constants depending on the solution and regularity bounds. The second estimate is conditional on dual regularity; corners or insufficient data regularity can reduce the rate.

The same Galerkin method handles evolution equations. For the homogeneous heat equation, the semidiscrete finite element heat equation is $M\dot{c}+Sc=0$. Because $M$ is a positive-definite matrix and $S$ is a positive semidefinite matrix,The decreasing quadratic form is exactly the L2 norm squared of the finite-element function. This separates spatial stability of a numerical method from the subsequent time integrator, just as in the Schrodinger equation example. Conformity, coercivity and approximation estimates together explain convergence.