Related papers: Unfitted boundary algebraic equation method based …
We present a spectrally-accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. Applications include computing the effective permeability of…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
A novel boundary element method (BEM) removes the classical dependence on explicit fundamental solutions and extends quasi-optimal BEM discretisations to strongly elliptic operators with variable coefficients. The approach constructs a…
This work presents a generalized boundary integral method for elliptic equations on surfaces, encompassing both boundary value and interface problems. The method is kernel-free, implying that the explicit analytical expression of the kernel…
We present a 3D hybrid method which combines the Finite Element Method (FEM) and the Spectral Boundary Integral method (SBIM) to model nonlinear problems in unbounded domains. The flexibility of FEM is used to model the complex,…
Elliptic partial differential equations (PDEs) are a major class of time-independent PDEs that play a key role in many scientific and engineering domains such as fluid dynamics, plasma physics, and solid mechanics. Recently, neural…
Boundary element methods (BEM) reduce a partial differential equation in a domain to an integral equation on the domain's boundary. They are particularly attractive for solving problems on unbounded domains, but handling the dense matrices…
A non-singular formulation of the boundary integral method (BIM) is presented for the Laplace equation whereby the well-known singularities that arise from the fundamental solution are eliminated analytically. A key advantage of this…
A second-order accurate kernel-free boundary integral method is presented for Stokes and Navier boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose…
This article presents an $O(N\log N)$ method for numerical solution of Maxwell's equations for dielectric scatterers using a 3D boundary integral equation (BIE) method. The underlying BIE method used is based on a hybrid…
We develop a family of stabilized backward differentiation formula (sBDF) schemes of orders one through four for semilinear parabolic equations. The proposed methods are designed to achieve three properties that are rarely available…
In this paper, we proposed two new types of edge multiscale methods motivated by \cite{GL18} to solve Partial Differential Equations (PDEs) with high-contrast heterogeneous coefficients: Edge spectral multiscale Finte Element method…
This paper proposes a new boundary integral equation (BIE) methodology based on the perfectly matched layer (PML) truncation technique for solving the electromagnetic scattering problems in a multi-layered medium. Instead of using the…
In this paper, we make the first attempt to apply the boundary integrated neural networks (BINNs) for the numerical solution of two-dimensional (2D) elastostatic and piezoelectric problems. BINNs combine artificial neural networks with the…
Conventionally, piecewise polynomials have been used in the boundary elements method (BEM) to approximate unknown boundary values. Since infinitely smooth radial basis functions (RBFs) are more stable and accurate than the polynomials for…
Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation…
Two combined methods for computing solutions of time-varying semilinear differential-algebraic equations (descriptor systems) are obtained. When constructing the methods, time-varying spectral projectors which can be found numerically are…
In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods…
We analyze the well posedness of certain field-only boundary integral equations (BIE) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the…