Session I.1 - Multiresolution and Adaptivity in Numerical PDEs
Poster
A posteriori error analysis of a linear Schrödinger type eigenvalue problem for atomic centered discretizations
Ioanna-Maria Lygatsika
Sorbonne Université, France - ioanna-maria.lygatsika@sorbonne-universite.fr
In this poster, we present a first a posteriori error analysis for variational approximations of the ground state eigenpair of a linear Schrödinger type eigenvalue problem for systems with one electron and M atoms, more precisely of the form Hu=λu, H=−Δ+∑Mi=1Vi+σ, ‖, with boundary conditions in one dimension. Denoting by (u_N,\lambda_N) the variational approximation of the ground state eigenpair (u,\lambda) based on a Gaussian discretization centered on atoms, we provide a posteriori estimates of the error in the energy norm \|u - u_N\|_H, when N goes to infinity. We introduce the residual of the equation and we decompose it into M residuals characterizing the error localized on atoms. It is shown that the bound can be expressed in terms of the dual "local" norms induced by the radially symmetric operators H_i=-\Delta + V_i + \sigma_i, i=1,\ldots,M centered on atoms. Such bound is fully computable as soon as an estimate on the dual local norms of the local residuals is available, which is obtained by performing a spectral decomposition of the bounded operators H_i of Hydrogen-like atoms. Finally, we provide numerical illustration of the performance of such a posteriori analysis on test cases.
Joint work with Mi-Song Dupuy (Sorbonne Université, France) and Geneviève Dusson (Université Bourgogne Franche-Comté, France).