Related papers: Geometrically Graded h-p Quadrature Applied to the…
Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed…
This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calder\'on calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density…
We introduce a two-phase approximation method designed to resolve singularities in three-dimensional harmonic Dirichlet problems. The approach utilizes the classical Green's function representation, decomposing the function into its…
Partial differential equations (PDEs) with Dirichlet boundary conditions defined on boundaries with simple geometry have been succesfuly treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of…
We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and…
We present a unified hard-constraint framework for solving geometrically complex PDEs with neural networks, where the most commonly used Dirichlet, Neumann, and Robin boundary conditions (BCs) are considered. Specifically, we first…
We study the problem of computing the conformal modulus of rings and quadrilaterals with strong singularities and cusps on their boundary. We reduce this problem to the numerical solution of the associated Dirichlet and Dirichlet-Neumann…
This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary H\"{o}lder regularity under proper…
We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions}, that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet-Neumann boundary conditions; of course, (slight…
Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted…
Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the…
This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition. The…
Many boundary element integral equation kernels are based on the Green's functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's equations.…
Although high-order Maxwell integral equation solvers provide significant advantages in terms of speed and accuracy over corresponding low-order integral methods, their performance significantly degrades in presence of non-smooth…
This article presents a new high-order accurate algorithm for finding a particular solution to a linear, constant-coefficient partial differential equation (PDE) by means of a convolution of the volumetric source function with the Green's…
The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) for a second order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develops the…
Physics-Informed Neural Networks (PINNs) solve partial differential equations (PDEs) by embedding governing equations and boundary/initial conditions into the loss function. However, enforcing Dirichlet boundary conditions accurately…
This paper presents an a priori error analysis of the hp-version of the boundary element method for the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. We use H(div)-conforming discretisations with…
The use of boundary integral equations in modeling boundary value problems-such as elastic, acoustic, or electromagnetic ones-is well established in the literature and widespread in practical applications. These equations are typically…
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…