The virtual element method for eigenvalue problems with potential terms on polytopal meshes
This provides a flexible numerical method for eigenvalue problems in quantum mechanics, particularly for Density Functional Theory, but the extension is incremental as it applies existing VEM techniques to a new problem class.
The paper extends the conforming virtual element method to solve eigenvalue problems with potential terms on polytopal meshes, achieving optimal convergence rates. Numerical tests on the Quantum Harmonic Oscillator and a singular eigenvalue problem confirm the method's accuracy.
We extend the conforming virtual element method to the numerical resolution of eigenvalue problems with potential terms on a polytopal mesh. An important application is that of the Schrodinger equation with a pseudopotential term. This model is a fundamental element in the numerical resolution of more complex problems from the Density Functional Theory. The VEM is based on the construction of the discrete bilinear forms of the variational formulation through certain polynomial projection operators that are directly computable from the degrees of freedom. The method shows a great flexibility with respect to the meshes and provide a correct spectral approximation with optimal convergence rates. This point is discussed from both the theoretical and the numerical viewpoint. The performance of the method is numerically investigated by solving the Quantum Harmonic Oscillator problem with the harmonic potential and a singular eigenvalue problem with zero potential for the first eigenvalues.