Exponentially convergent functional-discrete method for eigenvalue transmission problems with discontinuous flux and potential as a function in the space $L_1$
Provides a convergent numerical method for a class of eigenvalue problems with rough coefficients, but the contribution is incremental as it extends an existing technique to a new function space.
The paper develops a functional-discrete method for eigenvalue transmission problems with discontinuous flux and integrable potential, achieving superexponential convergence. Numerical examples confirm the theory and reveal analytical properties of eigensolutions for nonself-adjoint operators.
Based on the functional-discrete technique (FD-method), an algorithm for eigenvalue transmission problems with discontinuous flux and integrable potential is developed. The case of the potential as a function belonging to the functional space $L_1$ is studied for both linear and nonlinear eigenvalue problems. The sufficient conditions providing superexponential convergence rate of the method were obtained. Numerical examples are presented to support the theory. Based on the numerical examples and the convergence results, conclusion about analytical properties of eigensolutions for nonself-adjoint differential operators is made.