Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM

Let r = (x₁² + x₂²)^{1 / 2} be the distance function to the origin O ∈ R², and let us fix δ > 0. We consider the "Schrödinger-type mixed boundary value problem" - Δ u + δ r^{- 2} u = f ∈ H^{m - 1} (Ω) on a bounded polygonal domain Ω ⊂ R². The singularity in the potential δ r^{- 2} severely limits the regularity of the solution u. This affects the rate of convergence to u of the finite element approximations u_S ∈ S obtained using a quasi-uniform sequence of meshes. We show that a suitable graded sequence of meshes recovers the quasi-optimal convergence rate {norm of matrix} u - u_n {norm of matrix}_{H1 (Ω)} ≤ C dim (S_n)^{- m / 2} {norm of matrix} f {norm of matrix}_{Hm - 1 (Ω)}, where S_n are the FE spaces of continuous, piecewise polynomial functions of degree m ≥ 1 associated to our sequence of meshes and u_n = u_Sn ∈ S_n are the FE approximate solutions. This is in spite of the fact that u / ∈ H^{m + 1} (Ω) in general. One of the main results of our paper is to show that the singularities due to the potential and the singularities due to the singularities of the domain or to the change in boundary conditions can be treated in the same way. Our proof is based on regularity and well-posedness results in weighted Sobolev spaces, with the weight taking into account all singularities (including the ones coming from the potential). Our regularity results apply also to operators with weaker singularities, like the Schrödinger operator - Δ u + δ r^{- 1}, for which we also obtain Fredholm conditions and a formula for the index. Our a priori estimates also extend to piecewise smooth domains (i.e., curvilinear polygonal domains).