Related papers: Orbital-enriched Flat-top Partition of Unity Metho…
The current state of the art for large-scale quantum-mechanical simulations is the planewave (PW) pseudopotential method, as implemented in codes such as VASP, ABINIT, and many others. However, since the PW method uses a global Fourier…
This paper presents a conforming thin plate bending element based on the Partition of Unity Finite Element Method (PUFEM), for the simulation of steady-state forced vibration. The issue of ensuring the continuity of displacement and slope…
In this paper the $h$-adaptive partition-of-unity method and the $h$- and $hp$-adaptive finite element method are applied to partial differential equations arising in quantum mechanics, namely, the Schr\"odinger equation with Coulomb and…
Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Different physical models can exist within a PUM scheme for handling problems with zones of…
Accurate computation of multiple eigenvalues of quantum Hamiltonians is essential in quantum chemistry, materials science, and molecular spectroscopy. Estimating excited-state energies is challenging for classical algorithms due to…
Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Within the PUM global-local enrichment scheme [1, 2] different physical models can exist to…
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…
We investigate the scattering of scalar plane waves in two dimensions by a heterogeneous layer that is periodic in the direction parallel to its boundary. On describing the layer as a union of periodic laminae, we develop a solution of the…
The unified transform method (UTM) provides a novel approach to the analysis of initial-boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter…
We introduce an efficient algorithm, called partition of unity extension or PUX, to construct an extension of desired regularity of a function given on a complex multiply connected domain in $2D$. Function extension plays a fundamental role…
In this paper, we propose a hybrid method that combines finite element method (FEM) and physics-informed neural network (PINN) for solving linear elliptic problems. This method contains three steps: (1) train a PINN and obtain an…
Conventional finite element methods are known to be tedious in adaptive refinements due to their conformal regularity requirements. Further, the enrichment functions for adaptive refinements are often not readily available in general…
We consider a class of adaptive multilevel domain decomposition-like algorithms, built from a combination of adaptive multilevel finite element, domain decomposition, and partition of unity methods. These algorithms have several interesting…
We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite…
In recent years, an increasing attention has been paid to quantum heterostructures with tailored functionalities, such as heterojunctions and quantum matematerials, in which quantum dynamics of electrons can be described by the…
We propose an overlapping Schwarz space-time refinement framework for the material point method (OS-MPM) to improve computational efficiency in problems with strongly localized deformation, contact, and large geometric nonlinearity. The…
In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the $n$-dimensional hypersurface, $\Gamma…
We present a computationally efficient approach to perform large-scale all-electron density functional theory calculations by enriching the classical finite element basis with compactly supported atom-centered numerical basis functions that…
The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a large system of linear equations. Here we…
Our main objective in this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary…