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We analyze the first order Enlarged Enhancement Virtual Element Method (E2VEM) for the Poisson problem. The method allows the definition of bilinear forms that do not require a stabilization term, thanks to the exploitation of higher order polynomial projections that are made computable by suitably enlarging the enhancement property (from which comes the prefix E2) of local virtual spaces. We provide a sufficient condition for the well-posedness and optimal order a priori error estimates. We present numerical tests on convex and non-convex polygonal meshes that confirm the robustness of the method and the theoretical convergence rates. [ABSTRACT FROM AUTHOR]

We present a new high-order spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order hp-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series, which traditionally is used within the shifted boundary method. Here, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small cut cells. Dirichlet, Neumann, and Robin boundary conditions are enforced weakly through a generalized: (i) Nitsche’s method and (ii) Aubin’s method. A consistent asymptotic preserving formulation of the embedded Robin formulations is presented. Several experiments and analyses of the numerical properties of the various weak forms are showcased. We include convergence studies under polynomial increase of the basis functions, p, mesh refinement, h, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. With this, we assess the influence of errors across variational forms, polynomial order, mesh size, and mappings between the true and surrogate boundaries. [ABSTRACT FROM AUTHOR]

We investigate the concept of effective resistance in connection graphs, expanding its traditional application from undirected graphs. We propose a robust definition of effective resistance in connection graphs by focusing on the duality of Dirichlet-type and Poisson-type problems on connection graphs. Additionally, we delve into random walks, taking into account both node transitions and vector rotations. This approach introduces novel concepts of effective conductance and resistance matrices for connection graphs, capturing mean rotation matrices corresponding to random walk transitions. Thereby, it provides new theoretical insights for network analysis and optimization. [ABSTRACT FROM AUTHOR]

Symmetry properties of solutions to elliptic quasilinear equations have been widely studied in the context of Dirichlet boundary conditions. We show that, in the context of Robin boundary conditions, the symmetry property á la Gidas, Ni, and Nirenberg does not hold in dimension n ≥ 2, even for superharmonic functions, and we provide an explicit example. [ABSTRACT FROM AUTHOR]

We initiate the design and the analysis of stabilization-free Virtual Element Methods for the Poisson problem written in mixed form. A Virtual Element version of the lowest order Raviart-Thomas Finite Element is considered. To reduce the computational costs, a suitable projection on the gradients of harmonic polynomials is employed. A complete theoretical analysis of stability and convergence is developed in the case of quadrilateral meshes. Some numerical tests highlighting the actual behaviour of the scheme are also provided. [ABSTRACT FROM AUTHOR]

In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem $ -\Delta u = f(x, y) $ finding a geometric condition involving the curvature of the boundary and the normal derivative of $ f $ on the boundary to ensure uniqueness of the critical point. In the second part we consider stable solutions of the nonlinear problem $ -\Delta u = f(u) $ in perturbation of convex domains. [ABSTRACT FROM AUTHOR]

Our aim of this article is to study qualitative properties of Dirichlet problems involving the Hardy-Leray operator ℒV≔−Δ+V{{\mathcal{ {\mathcal L} }}}_{V}:= -\Delta +V, where V(x)=∑i=1mμi∣x−Ai∣2V\left(x)={\sum }_{i=1}^{m}\frac{{\mu }_{i}}{{| x-{A}_{i}| }^{2}}, with μi≥−(N−2)24{\mu }_{i}\ge -\frac{{\left(N-2)}^{2}}{4} being the Hardy-Leray potential containing the polars’ set Am={Ai:i=1,…,m}{{\mathcal{A}}}_{m}=\left\{{A}_{i}:i=1,\ldots ,m\right\} in RN{{\mathbb{R}}}^{N} (N≥2N\ge 2). Since the inverse-square potentials are critical with respect to the Laplacian operator, the coefficients {μi}i=1m{\left\{{\mu }_{i}\right\}}_{i=1}^{m} and the locations of polars {Ai}\left\{{A}_{i}\right\} play an important role in the properties of solutions to the related Poisson problems subject to zero Dirichlet boundary conditions. Let Ω\Omega be a bounded domain containing Am{{\mathcal{A}}}_{m}. First, we obtain increasing Dirichlet eigenvalues: ℒVu=λuinΩ,u=0on∂Ω,{{\mathcal{ {\mathcal L} }}}_{V}u=\lambda u\hspace{1.0em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1.0em}{\rm{on}}\hspace{0.33em}\partial \Omega , and the positivity of the principle eigenvalue depends on the strength μi{\mu }_{i} and polars’ setting. When the spectral does not contain the origin, we then consider the weak solutions of the Poisson problem (E)ℒVu=νinΩ,u=0on∂Ω,\left(E)\hspace{1.0em}\hspace{1.0em}{{\mathcal{ {\mathcal L} }}}_{V}u=\nu \hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1em}{\rm{on}}\hspace{0.33em}\partial \Omega , when ν\nu belongs to Lp(Ω){L}^{p}\left(\Omega ), with p>2NN+2p\gt \frac{2N}{N+2} in the variational framework, and we obtain a global weighted L∞{L}^{\infty } estimate when p>N2p\gt \frac{N}{2}. When the principle eigenvalue is positive and ν\nu is a Radon measure, we build a weighted distributional framework to show the existence of weak solutions of problem (E)\left(E). Moreover, via this weighted distributional framework, we can obtain a sharp assumption of ν∈Cγ(Ω¯\Am)\nu \in {{\mathcal{C}}}^{\gamma }\left(\bar{\Omega }\setminus {{\mathcal{A}}}_{m}) for the existence of isolated singular solutions for problem (E)\left(E).

This paper is devoted to the study of the existence and the uniqueness of the solution of the Stochastic Dirichlet-Poisson problems on Hilbert spaces. We prove that the unique solution of this equation is given by a probabilistic formula. [ABSTRACT FROM AUTHOR]

A neural-network-assisted numerical method is proposed for the solution of Laplace and Poisson problems. Finite differences are applied to approximate the spatial Laplacian operator on nonuniform grids. For this, a neural network is trained to compute the corresponding coefficients for general quadrilateral meshes. Depending on the position of a given grid point x 0 and its neighbors, we face with a nonlinear optimization problem to obtain the finite difference coefficients in x 0 . This computing step is executed with an artificial neural network. In this way, for any geometric setup of the neighboring grid points, we immediately obtain the corresponding coefficients. The construction of an appropriate training data set is also discussed, which is based on the solution of overdetermined linear systems. The method was experimentally validated on a number of numerical tests. As expected, it delivers a fast and reliable algorithm for solving Poisson problems. [ABSTRACT FROM AUTHOR]

A neural-network-assisted numerical method is proposed for the solution of Laplace and Poisson problems. Finite differences are applied to approximate the spatial Laplacian operator on nonuniform grids. For this, a neural network is trained to compute the corresponding coefficients for general quadrilateral meshes. Depending on the position of a given grid point x0 and its neighbors, we face with a nonlinear optimization problem to obtain the finite difference coefficients in x0. This computing step is executed with an artificial neural network. In this way, for any geometric setup of the neighboring grid points, we immediately obtain the corresponding coefficients. The construction of an appropriate training data set is also discussed, which is based on the solution of overdetermined linear systems. The method was experimentally validated on a number of numerical tests. As expected, it delivers a fast and reliable algorithm for solving Poisson problems.

Then the coefficients HT ht Graph are admissible for constructing WEB-Splines according to Definition 2.1. Now consider HT ht , I j i I K i , Eqs. [Extracted from the article]

Given a sequence of finite element spaces which form a de Rham sequence, we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The dual representations also need to satisfy the de Rham sequence on the domain boundary. The matrix which converts primal representations to dual representations – the Hodge matrix – is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even for high order methods. The differential operators for dual representations will be defined. Vector operations, grad, curl and div, for primal and dual representations are both topological and do not depend on the metric, i.e. the size and shape of the mesh or the order of the numerical method. Derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations. As illustration of the use of dual representations, the method will be applied to (i) a mixed formulation for the Poisson problem in 3D, (ii) it will be shown that this approach allows one to preserve the equivalence between Dirichlet and Neumann problems in the finite dimensional setting and, (iii) the method will be applied to the approximation of grad–div eigenvalue problem on affine and non-affine meshes. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a Stabilization Free Virtual Element Method (SFVEM), that allows the definition of bilinear forms that do not require an arbitrary stabilization term, thanks to the exploitation of higher-order polynomial projections on divergence free vectors of polynomials. The method is introduced in the lowest order formulation for the Poisson problem. We provide a sufficient condition on the polynomial projection space that implies the well-posedness, proved on particular classes of polygons, and optimal order a priori error estimates. Numerical tests on convex and non-convex polygonal meshes confirm the theoretical convergence rates and show that the method is suitable for solving problems characterized by anisotropies. • Virtual Element method without an arbitrary stabilization. • Structure preserving bilinear form. • Well-posedness analysis. • Robustness with respect to anisotropic coefficients and solutions. [ABSTRACT FROM AUTHOR]

We use a three‐field mixed formulation of the Poisson equation to develop a mixed finite element method using Raviart–Thomas elements. We use a locally constructed biorthogonal system for Raviart–Thomas finite elements to improve the computational efficiency of the approach. We analyze the existence, uniqueness and stability of the discrete problem and show an a priori error estimate. We also develop an a posteriori error estimate for our formulation. Numerical results are presented to demonstrate the performance of our approach. [ABSTRACT FROM AUTHOR]

Given a continuous function on the boundary of a bounded open set in R d there exists a unique bounded harmonic function, called the Perron solution, taking the prescribed boundary values at least at all regular points (in the sense of Wiener) of the boundary. We extend this result to vector-valued functions and consider several methods of constructing the Perron solution which are classical in the real-valued case. Special emphasis is on the case where the codomain is a Banach lattice. In this case we investigate Perron's classical construction via sub- and supersolutions. We also apply our results to solve elliptic and parabolic boundary value problems of vector-valued functions. [ABSTRACT FROM AUTHOR]

We treat three problems on a two-dimensional "punctured periodic domain": we take Ω r = (− L , L) 2 \ r K , where r > 0 and K is the closure of an open connected set that is star-shaped with respect to 0 and has a C 1 boundary. We impose periodic boundary conditions on the boundary of Ω = (− L , L) 2 , and Dirichlet boundary conditions on ∂ (r K). In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier–Stokes equations, all with a fixed forcing function f , and examine the behavior of solutions as r → 0. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of Ω with periodic boundary conditions. [ABSTRACT FROM AUTHOR]

A new finite element method/boundary element method (FEM/BEM) scheme is proposed for the solution of the 2D magnetic static and quasi-static problems with unbounded domains. The novelty is an original approach in the treatment of the outer region. The related domain integral is eliminated at the discrete level by using the finite element approximation of the fundamental solutions (Green’s functions) at every node of the related mesh. This “FEM-Green” approach replaces the standard boundary element method. It is simpler to implement because no integration on the boundary of the domain is required. Then, the method leads to a substantially reduced computational burden. Moreover, the coupling with finite elements is more natural since it is based on the same Galerkin approximation. Some examples with open boundary and nonlinear materials are presented and compared with the standard finite element method.

We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source belongs to the dual of a weighted Sobolev space where the weight belongs to the Muckenhoupt class. Second, we prove the stability in weighted norms for standard finite element approximations under the quasi-uniformity assumption on the family of meshes. [ABSTRACT FROM AUTHOR]

We propose and validate a novel extension of Hybrid High-Order (HHO) methods to meshes featuring curved elements. HHO methods are based on discrete unknowns that are broken polynomials on the mesh and its skeleton. We propose here the use of physical frame polynomials over mesh elements and reference frame polynomials over mesh faces. With this choice, the degree of face unknowns must be suitably selected in order to recover on curved meshes the same convergence rates as on straight meshes. We provide an estimate of the optimal face polynomial degree depending on the element polynomial degree and on the so-called effective mapping order. The estimate is numerically validated through specifically crafted numerical tests. All test cases are conducted considering two- and three-dimensional pure diffusion problems, and include comparisons with discontinuous Galerkin discretizations. The extension to agglomerated meshes with curved boundaries is also considered. [ABSTRACT FROM AUTHOR]

We present an analytical study of discretization stencils for the Poisson problem and the incompressible Navier–Stokes problem when used with some direct forcing immersed boundary methods. This study uses, but is not limited to, second-order discretization and Ghost-Cell Finite-Difference methods. We show that the stencil size increases with the aspect ratio of rectangular cells, which is undesirable as it breaks assumptions of some linear system solvers. To circumvent this drawback, a modification of the Ghost-Cell Finite-Difference methods is proposed to reduce the size of the discretization stencil to the one observed for square cells, i.e. with an aspect ratio equal to one. Numerical results validate this proposed method in terms of accuracy and convergence, for the Poisson problem and both Dirichlet and Neumann boundary conditions. An improvement on error levels is also observed. In addition, we show that the application of the chosen Ghost-Cell Finite-Difference methods to the Navier–Stokes problem, discretized by a pressure-correction method, requires an additional interpolation step. This extra step is implemented and validated through well known test cases of the Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

We consider a model Poisson problem in ℝd (d=2,3) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges (d=2) or small faces (d=3). Our results extend the ones in [L. Beirão da Veiga, C. Lovadina and A. Russo, Stability analysis for the virtual element method, Math. Models Methods Appl. Sci.27 (2017) 2557–2594] for the original two-dimensional virtual element method from [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci.23 (2013) 199–214] to the version of the virtual element method in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl.66 (2013) 376–391] that can also be applied to problems in three dimensions. [ABSTRACT FROM AUTHOR]

A fictitious domain approach for the solution of second-order linear differential problems is proposed; spectral/hp elements have been used for the discretization of the domain. The peculiarity of our approach is that the Lagrange multipliers are particular distributed functions, instead of classical δ Dirac (impulsive) multipliers. In this paper we present the formulation and the application of this approach to 1D and 2D Poisson problems and 2D Stokes flow (biharmonic equation). [ABSTRACT FROM AUTHOR]