Past exam of the numerical solutions of partial differential equations course of the University of Oxford / 2023 / 2 / a

For $u^{(m,n)}(x,y)=\sin (m\pi x)\sin (n\pi y)$, direct differentiation yieldsSince the planar Laplacian is $\Delta ={\partial }_{xx}+{\partial }_{yy}$, the required eigenvalue of $-\Delta +I$ isOn the sides $x=0,1$ the first sine factor vanishes, and on $y=0,1$ the second sine factor vanishes. Thus the homogeneous Dirichlet boundary condition holds on the whole boundary. The eigenfunction is nonzero, for instance at $(x,y)=(1/(2m),1/(2n))$, where its value is one.