Related papers: A multiscale finite element method for the Schr\"{…
We present in this paper a rather general method for the construction of so-called conditionally exactly solvable potentials. This method is based on algebraic tools known from supersymmetric quantum mechanics. Various families of…
New low-order $H(\textrm{div})$-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the $(d+1)$-order…
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{\"o}dinger equation in the semiclassical limit. We specifically analyse the…
In this paper, we construct a combined multiscale finite element method (MsFEM) using the Local Orthogonal Decomposition (LOD) technique to solve the multiscale problems which may have singularities in some special portions of the…
In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: \begin{equation*} \left\{\begin{array}{ll} -\Delta u_j + au_j = \mu_ju_j^3+\beta\sum_{k\ne j}u_k^2u_j, u_j>0\ \ \hbox{in}\ \Omega, u_j=0 \…
We present a semidefinite program (SDP) algorithm to find eigenvalues of Schr\"{o}dinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the…
We focus on a recently developed generalized pseudospectral method for accurate, efficient treatment of certain central potentials of interest in various branches in quantum mechanics, usually having singularity. Essentially this allows…
We prove an explicit weighted estimate for the semiclassical Schr\"odinger operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$ a finite signed measure, and where $h >0$ is the semiclassical parameter. The proof is…
In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method and the Galerkin finite element method are used to discretize the model…
We present a method for efficiently finding solutions of L\"uscher's quantisation condition, the equation which relates two-particle scattering amplitudes to the discrete spectrum of states in a periodic spatial volume of finite extent such…
The quantum behavior of charge carriers in semiconductor structures is often described in terms of the effective mass Schr\"{o}dinger equation, neglecting the rapid fluctuations of the wave function on the scale of the atomic lattice. For…
In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and error estimates of the finite element solution…
We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE.…
Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the…
We introduce an integrated meshing and finite element method pipeline enabling black-box solution of partial differential equations in the volume enclosed by a boundary representation. We construct a hybrid hexahedral-dominant mesh, which…
In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Unlike the biharmonic…
In the present work, we investigate the computational efficiency afforded by higher-order finite-element discretization of the saddle-point formulation of orbital-free density functional theory. We first investigate the robustness of viable…
This paper concerns the preconditioning technique for discrete systems arising from time-harmonic Maxwell equations with absorptions, where the discrete systems are generated by N\'ed\'elec finite element methods of fixed order on meshes…
Quantum mechanical calculations require the repeated solution of a Schr\"odinger equation for the wavefunctions of the system. Recent work has shown that enriched finite element methods significantly reduce the degrees of freedom required…
Based on the fact that the Hamiltonians of the Coulomb many-particle systems are always factorized we develop the two different approaches for analytical solution of the Schr\"{o}dinger equation written for arbitrary few- and many-particle…