Related papers: Cross-points in the Dirichlet-Neumann method II: a…
Domain decomposition (DD) methods for solving time-dependent problems can be classified by (i) the method of domain decomposition used, (ii) the choice of decomposition operators (exchange of boundary conditions), and (iii) the splitting…
We present a Waveform Relaxation (WR) version of the Dirichlet-Neumann algorithm, formulated specially for multiple subdomains splitting for general parabolic and hyperbolic problems. This method is based on a non-overlapping spatial domain…
This paper presents four novel domain decomposition algorithms integrated with nonlinear mapping techniques to address collocation-based solutions of eigenvalue problems involving sharp interfaces or steep gradients. The proposed methods…
We propose a simple domain decomposition method for $d$-dimensional elliptic PDEs which involves an overlapping decomposition into local subdomain problems and a global coarse problem. It relies on a space-filling curve to create equally…
An iteration-free method of domain decomposition is considered for approximate solving a boundary value problem for a second-order parabolic equation. A standard approach to constructing domain decomposition schemes is based on a partition…
We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We…
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
For scalar fully nonlinear partial differential equations depending on the Hessian andspatial coordinates, we present a general theory for obtaining comparison principles and well posedness for the associated Dirichlet problem with…
The elliptic system of equations, which is general-covariant and locally SU(2)-covariant, is investigated. The new condition of the Dirichlet problem solvability and the condition of zeros absence for solutions are obtained for this system,…
This paper studies the $d$-dimensional extension of a fictitious domain penalization technique that we previously proposed for Neumann or Robin boundary conditions. We apply Droniou's approach for non-coercive linear elliptic problems to…
We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices,…
In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination…
The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of…
A rigorous mathematical framework is provided for a substructuring-based domain-decomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a…
We consider the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the…
We study the Dirichlet problem for a class of curvature equations arising from conformal geometry on Riemannian manifolds $(M^n, g)$ with boundary where $n \geq 3$. We prove there exists a unique solution using the continuity method which…
In this paper, based on the overlapping domain decomposition method (DDM) proposed in \cite{Leng2015}, an one step preconditioner is proposed to solve 2D high frequency Helmholtz equation. The computation domain is decomposed in both $x$…
The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a…
In the present paper we describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Neumann boundary conditions. It is well known that in such cases the solutions have singularities near the corners which…
Many tasks in geometry processing are modeled as variational problems solved numerically using the finite element method. For solid shapes, this requires a volumetric discretization, such as a boundary conforming tetrahedral mesh.…