Related papers: Geometrically Graded h-p Quadrature Applied to the…
Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the…
In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original…
In recent work, Li et al.\ (Comm.\ Math.\ Sci., 7:81-107, 2009) developed a diffuse-domain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The…
We study a discrete analogue of the parametric Plateau problem in a non-polynomial tensor-product surface spaces generated by the generalized trigonometric (GT)--B\'ezier basis. Boundary interpolation is imposed by prescribing the boundary…
This paper is concerned with approximations and related discretization error estimates for the normal derivatives of solutions of linear elliptic partial differential equations. In order to illustrate the ideas, we consider the Poisson…
The Kernel-Free Boundary Integral (KFBI) method presents an iterative solution to boundary integral equations arising from elliptic partial differential equations (PDEs). This method effectively addresses elliptic PDEs on irregular domains,…
This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for…
In this article, the authors survey and review the studies of boundary value problems for regular functions in Clifford analysis, which include theoretical foundations and useful methods. Its theoretical bases consist of the generalized…
We consider a discontinuous Galerkin method for the numerical solution of boundary value problems in two-dimensional domains with curved boundaries. A key challenge in this setting is the potential loss of convergence order due to…
This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions for which Dirichlet and Neumann conditions are specified on…
Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in K-dimensional space such that the…
In this paper, we prove that there exists a unique weak solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is…
The solution of partial differential equations (PDEs) on complex domains often presents a significant computational challenge by requiring the generation of fitted meshes. The Diffuse Domain Method (DDM) is an alternative which reformulates…
An hp-version interior penalty discontinuous Galerkin (IPDG) method under nonconforming meshes is proposed to solve the quad-curl eigenvalue problem. We prove well-posedness of the numerical scheme for the quad-curl equation and then derive…
We propose a method combining boundary integral equations and neural networks (BINet) to solve partial differential equations (PDEs) in both bounded and unbounded domains. Unlike existing solutions that directly operate over original PDEs,…
Obtaining high-precision guaranteed lower eigenvalue bounds remains difficult, even though the standard high-order conforming finite element (FEM) easily yields extremely sharp upper bounds. Recently developed rigorous approaches using such…
We investigate error bounds for numerical solutions of divergence structure linear elliptic PDEs on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of…
This paper is concerned with minimization of a fourth-order linearized Canham-Helfrich energy subject to Dirichlet boundary conditions on curves inside the domain. Such problems arise in the modeling of the mechanical interaction of…
A new gridding technique for the solution of partial differential equations in cubical geometry is presented. The method is based on volume penalization, allowing for the imposition of a cubical geometry inside of its circumscribing sphere.…
A system of boundary-domain integral equations is derived from the bidimensional Dirichlet problem for the diffusion equation with variable coefficient using the novel parametrix from [22] different from the one in [5,18]. Mapping…