I am studying the discretization of Poisson's equation in $1D$. In Matlab I created different discretization matrices (Laplace operator) according to different sizes of the mesh:% ParametersN = [40, 80, 160, 1280]; % Dimension 1-D grid% Laplace operatorsL1 = -1/((1/N(1))^2) * full( ( spdiags(repmat([1,-2,1], N(1)-1, 1), -1:1, N(1)-1, N(1)-1) ) );L2 = -1/((1/N(2))^2) * full( ( spdiags(repmat([1,-2,1], N(2)-1, 1), -1:1, N(2)-1, N(2)-1) ) );L3 = -1/((1/N(3))^2) * full( ( spdiags(repmat([1,-2,1], N(3)-1, 1), -1:1, N(3)-1, N(3)-1) ) );L4 = -1/((1/N(4))^2) * full( ( spdiags(repmat([1,-2,1], N(4)-1, 1), -1:1, N(4)-1, N(4)-1) ) );I then plotted the eigenvalues of the matrices using[V1, D1] = eig(L1);plot(diag(D1) / norm(diag(D1),2), '--');What I see is that as I increase $N$ (so the step size goes to zero), the difference between the biggest and the smallest eigenvalues decreases. Therefore the condition number is getting better. Is there anything else I can see from such a plot?
Spectrum of the Laplace operator
matlab;eigenvalues;poisson