Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators (Q1904150)

The homotopy method is used to compute the smallest few eigenvalues and their corresponding eigenfunctions of a linear self-adjoint partial differential operator. By this method the homotopy \(H(t) = (1 - t)A_0 + tA_1\) is formed (\(A_0\) is a matrix whose smallest few eigenpairs are known). The eigenpairs at \(t + dt\) are obtained using the Rayleigh quotient iterations, \(dt\) is chosen so that \(H(t)\) and \(H(t + dt)\) do not differ too much. This step is repeated until \(t\) reaches 1, when the solution to the original problem is found. The issues relating to the linear solver and the low-rank perturbations and the case when several eigenvalues are clustered together are discussed. Numerical examples for the Schrödinger eigenvalue problem in two space dimensions are given. The algorithm is used to locate the bifurcation point of a parametrized partial differential equation.