Related papers: A numerical domain decomposition method for solvin…
Recent developments in mechanical, aerospace, and structural engineering have driven a growing need for efficient ways to model and analyse structures at much larger and more complex scales than before. While established numerical methods…
he Singular Manifold Method is presented as an excellent tool to study a 2+1 dimensional equation in despite of the fact that the same method presents several problems when applied to 1+1 reductions of the same equation. Nevertheless these…
A novel numerical approach to solving the shallow-water equations on the sphere using high-order numerical discretizations in both space and time is proposed. A space-time tensor formalism is used to express the equations of motion…
The goal of this paper is to study how the symmetry of the spherical domain influences solutions of elliptic equations on such domain. The method pursued is a variant of the moving plane method, discovered by Alexandrov (1962) and used for…
This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new…
We establish a rigidity theorem for annular sector-like domains in the setting of overdetermined elliptic problems on model Riemannian manifolds. Specifically, if such a domain admits a solution to the inhomogeneous Helmholtz equation…
The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…
This paper proposes a model order reduction method for a class of parametric dynamical systems. Using a temporal Fourier transform, we reformulate these systems into complex-valued elliptic equations in the frequency domain, containing…
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans…
We prove linear convergence for a new family of modified Dirichlet--Neumann methods applied to quasilinear parabolic equations, as well as the convergence of the Robin--Robin method. Such nonoverlapping domain decomposition methods are…
In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on…
A method is described by which a function defined on a cubic grid (as from a finite difference solution of a partial differential equation) can be resolved into spherical harmonic components at some fixed radius. This has applications to…
Solving elliptic PDEs in more than one dimension can be a computationally expensive task. For some applications characterised by a high degree of anisotropy in the coefficients of the elliptic operator, such that the term with the highest…
Two-level domain decomposition (DD) methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level…
This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated…
We investigate the performance of algebraic multigrid methods for the solution of the linear system of equations arising from a Virtual Element discretization. We provide numerical experiments on very general polygonal meshes for a model…
We study questions of existence and uniqueness of quadrature domains using computational tools from real algebraic geometry. These problems are transformed into questions about the number of solutions to an associated real semi-algebraic…
Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient…
A new domain decomposition method is introduced for the heterogeneous 2-D and 3-D Helmholtz equations. Transmission conditions based on the perfectly matched layer (PML) are derived that avoid artificial reflections and match incoming and…
We generalize the technique of [Solving Dirichlet boundary-value problems on curved domains by extensions from subdomains, SIAM J. Sci. Comput. 34, pp. A497--A519 (2012)] to elliptic problems with mixed boundary conditions and elliptic…