Related papers: An optimal complexity spectral solver for the Pois…
We present a numerical method for solving the Poisson equation on a nested grid. The nested grid consists of uniform grids having different grid spacing and is designed to cover the space closer to the center with a finer grid. Thus our…
We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones $K\subset \mathbb{R}^3$ in the specific scale $\ B^{\alpha}_{\tau,\tau}, \…
We develop a universally applicable embedded boundary finite difference method, which results in a symmetric positive definite linear system and does not suffer from small cell stiffness. Our discretization is efficient for the wave, heat…
Pseudospectral time domain (PSTD) methods are widely used in many branches of acoustics for the numerical solution of the wave equation, including biomedical ultrasound and seismology. The use of the Fourier collocation spectral method in…
We introduce an efficient boundary-adapted spectral method for peridynamic diffusion problems with arbitrary boundary conditions. The spectral approach transforms the convolution integral in the peridynamic formulation into a multiplication…
The Poisson-Boltzmann equation offers an efficient way to study electrostatics in molecular settings. Its numerical solution with the boundary element method is widely used, as the complicated molecular surface is accurately represented by…
We present a novel spectral method for the Allen-Cahn equation on spheres, eliminating the reliance on conventional quadrature exactness conditions. By replacing these conditions with a restricted isometry relation derived from…
The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes…
We obtain an algorithm computing explicitly the values of the non solvable spectral radii of convergence of the solutions of a differential module over a point of type 2, 3 or 4 of the Berkovich affine line.
A method for approximating sixth-order ordinary differential equations is proposed, which utilizes a deep learning feedforward artificial neural network, referred to as a neural solver. The efficacy of this unsupervised machine learning…
The goal of the present work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain. In this setting, transparent boundary conditions are vital to allow waves to leave (or even re-enter) the,…
Based on the Fourier extension, we propose an oversampling collocation method for solving the elliptic partial differential equations with variable coefficients over arbitrary irregular domains. This method only uses the function values on…
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into…
In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, ``[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless…
We present a unified algorithmic framework for quantum simulation of non-unitary dynamics and matrix functions, governed by the principle of spectral aliasing derived from the Poisson Summation Formula (PSF). By reinterpreting…
Polynomial spectral methods produce fast, accurate, and flexible solvers for broad ranges of PDEs with one bounded dimension, where the incorporation of general boundary conditions is well understood. However, automating extensions to…
An essential ingredient of a spectral method is the choice of suitable bases for test and trial spaces. On complex domains, these bases are harder to devise, necessitating the use of domain partitioning techniques such as the spectral…
Efficiently solving Poisson equations on complex, irregular domains remains a fundamental challenge in scientific computing, as classical iterative solvers often suffer from prohibitive runtime due to ill-conditioned systems. While neural…
We describe an accelerated direct solver for the integral equations which model acoustic scattering from curved surfaces. Surfaces are specified via a collection of smooth parameterizations given on triangles, a setting which generalizes…
An efficient numerical method for computing angle-resolved light scattering off periodic arrays is presented. The method combines finite-element discretization with a Schur complement solver. A significant speed-up of the computations in…