High-Precision Calculation of the Eigenvalues of the Laplace Operator

An eigenvalue problem for the Laplace operator is considered by assuming smooth functions that are complex-values and self-adjoint. The discretization of the two-dimensional problem reduces to that of one-dimensional problems with known solutions, which is convenient for testing the method. The interpolation formula is applied to the function of Laplace operator and the interpolation error is found. The discretization error is discarded to obtain eigenvalue problem depending on the vector of appropriate values of the required eigenfunctions at the grid points. A grid with various points is sufficient for obtaining the first eigenvalue and the computations on a grid give eigenvalues. The results show that the obtained eigenvalue results are sufficient foe constructing a discretization of the two-dimensional problem.