Laplace's Equation — Question 10

Question 10 – Solution

Strategy. Convert a bounded residual into an explicit comparison barrier, while retaining a separate bound for incorrect boundary values.

Step 1: Construct the barrier. Set $$\boxed{\psi(x,y)=\frac{x(1-x)+y(1-y)}4.}$$ It is nonnegative on the closed square and satisfies $\Delta\psi=-1$. Each quadratic is at most $1/4$, so $\max\psi=1/8$ at $(1/2,1/2)$. The barrier need not vanish on every edge; its nonnegative boundary values are sufficient for the comparison.

Step 2: Compare the error in both directions. Put $e=u-v$, so $\Delta e=-\Delta v$. For $w_+=e-\varepsilon-\delta\psi$ and $w_-=-e-\varepsilon-\delta\psi$, $$\Delta w_+=-\Delta v+\delta\ge 0,\qquad \Delta w_-=\Delta v+\delta\ge 0.$$ On the boundary, $|e|=|g-v|\le\varepsilon$ and $\psi\ge 0$, so both $w_\pm\le 0$. The stated maximum principle yields $$\boxed{|u-v|\le\varepsilon+\delta\psi(x,y)\le\varepsilon+\delta/8.}$$ This is an analytic bound over the entire square, not an inference from sampled residuals.

Step 3: Compare the certificate with the exact error. The exact harmonic solution for $g=x+y$ is $u=x+y$. The added factor vanishes on every edge, so $\varepsilon=0$. Direct differentiation gives $$\Delta v=-2a[x(1-x)+y(1-y)],\qquad \|\Delta v\|_\infty=|a|.$$ Thus the certificate gives $\|u-v\|_\infty\le |a|/8$. The exact error is $|a|x(1-x)y(1-y)$, with maximum $|a|/16$ at the center. The certificate is valid and overestimates this example’s maximum by a factor of two.

Step 4: Identify what must be checked and what is sharp. The field $v=u+M$ has zero residual for every constant $M$, yet its error is $|M|$ everywhere. Without a boundary-error check, an arbitrarily inaccurate temperature can pass the PDE test. Taking $|M|=\varepsilon$ and $\delta=0$ attains the bound $\|u-v\|_\infty=\varepsilon$, proving that coefficient one is sharp. The polynomial example does not attain the residual coefficient $1/8$ and does not prove it sharp. The barrier supplies a rigorous convenient certificate without a claim of optimality.