High-order Galerkin method for Helmholtz and Laplace problems on multiple open arcs

2020 ◽  
Vol 54 (6) ◽  
pp. 1975-2009
Author(s):  
Carlos Jerez-Hanckes ◽  
José Pinto
Keyword(s):  
Dirichlet Boundary ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Dimensional Space ◽  
Boundary Integral ◽  
Dirichlet Boundary Conditions ◽  
Boundary Integral Method ◽  
Finite Collection ◽  
Variational Form ◽  
Discrete Problems

We present a spectral Galerkin numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on a finite collection of open arcs in two-dimensional space. A boundary integral method is employed, giving rise to a first kind Fredholm equation whose variational form is discretized using weighted Chebyshev polynomials. Well-posedness of the discrete problems is established as well as algebraic or even exponential convergence rates depending on the regularities of both arcs and excitations. Our numerical experiments show the robustness of the method with respect to number of arcs and large wavenumber range. Moreover, we present a suitable compression algorithm that further accelerates computational times.

scholarly journals Fractional order elliptic problems with inhomogeneous Dirichlet boundary conditions

2020 ◽  
Vol 23 (2) ◽  
pp. 378-389
Author(s):  
Ferenc Izsák ◽  
Gábor Maros
Keyword(s):  
Fractional Order ◽  
Boundary Data ◽  
Dirichlet Boundary ◽  
Elliptic Problems ◽  
Integral Form ◽  
Uniqueness Result ◽  
Boundary Integral ◽  
Dirichlet Boundary Conditions ◽  
Two Dimensional ◽  
Potential Operators

AbstractFractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposed as a suitable mathematical model. The corresponding theory is completed by sharpening the mapping properties of the corresponding potential operators. The existence-uniqueness result is stated also for two-dimensional domains. Finally, a mild condition is provided to ensure the existence of the classical solution of the boundary integral equation.

Checking the Automated Construction of Finite Element Simulations From Dirichlet Boundary Conditions

2019 ◽  
Author(s):  
Kevin N. Chiu ◽  
Mark D. Fuge
Keyword(s):  
Finite Element ◽  
Topology Optimization ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Fem Simulation ◽  
Computational Design ◽  
Dirichlet Boundary Conditions ◽  
Governing Equations ◽  
Variational Form ◽  
Key Issues

Abstract From engineering analysis and topology optimization to generative design and machine learning, many modern computational design approaches require either large amounts of data or a method to generate that data. This paper addresses key issues with automatically generating such data through automating the construction of Finite Element Method (FEM) simulations from Dirichlet boundary conditions. Most past work on automating FEM assumes prior knowledge of the physics to be run or is limited to a small number of governing equations. In contrast, we propose three improvements to current methods of automating the FEM: (1) completeness labels that guarantee viability of a simulation under specific conditions, (2) type-based labels for solution fields that robustly generate and identify solution fields, and (3) type-based labels for variational forms of governing equations that map the three components of a simulation set — specifically, boundary conditions, solution fields, and a variational form — to each other to form a viable FEM simulation. We implement these improvements using the FEniCS library as an example case. We show that our improvements increase the percent of viable simulations that are run automatically from a given list of boundary conditions. This paper’s procedures ultimately allow for the automatic — i.e., fully computer-controlled — construction of FEM multi-physics simulations and data collection required to run data-driven models of physics phenomena or automate the exploration of topology optimization under many physics.

scholarly journals Application of Energetic BEM to 2D Elastodynamic Soft Scattering Problems

2019 ◽  
Vol 10 (1) ◽  
pp. 182-198
Author(s):  
A. Aimi ◽  
L. Desiderio ◽  
M. Diligenti ◽  
C. Guardasoni
Keyword(s):  
Wave Propagation ◽  
Dirichlet Boundary ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Dirichlet Boundary Conditions ◽  
Benchmark Problems ◽  
Weak Formulation ◽  
Scattering Problems ◽  
Propagation Analysis ◽  
First Time

Abstract Starting from a recently developed energetic space-time weak formulation of the Boundary Integral Equations related to scalar wave propagation problems, in this paper we focus for the first time on the 2D elastodynamic extension of the above wave propagation analysis. In particular, we consider elastodynamic scattering problems by open arcs, with vanishing initial and Dirichlet boundary conditions and we assess the efficiency and accuracy of the proposed method, on the basis of numerical results obtained for benchmark problems having available analytical solution.

Exponential convergence for hp-version and spectral finite element methods for elliptic problems in polyhedra

2015 ◽  
Vol 25 (09) ◽  
pp. 1617-1661 ◽  
Author(s):  
Dominik Schötzau ◽  
Christoph Schwab
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Elliptic Boundary ◽  
Building Blocks ◽  
Dirichlet Boundary Conditions ◽  
Patch Type ◽  
Polynomial Degree ◽  
Spectral Finite Element

We establish exponential convergence of conforming hp-version and spectral finite element methods for second-order, elliptic boundary-value problems with constant coefficients and homogeneous Dirichlet boundary conditions in bounded, axiparallel polyhedra. The source terms are assumed to be piecewise analytic. The conforming hp-approximations are based on σ-geometric meshes of mapped, possibly anisotropic hexahedra and on the uniform and isotropic polynomial degree p ≥ 1. The principal new results are the construction of conforming, patchwise hp-interpolation operators in edge, corner and corner-edge patches which are the three basic building blocks of geometric meshes. In particular, we prove, for each patch type, exponential convergence rates for the H1-norm of the corresponding hp-version (quasi)interpolation errors for functions which belong to a suitable, countably normed space on the patches. The present work extends recent hp-version discontinuous Galerkin approaches to conforming Galerkin finite element methods.

scholarly journals JACOBI ELLIPTIC SOLUTIONS OF λϕ4 THEORY IN A FINITE DOMAIN

2000 ◽  
Vol 15 (17) ◽  
pp. 2645-2659 ◽  
Author(s):  
J. A. ESPICHÁN CARRILLO ◽  
A. MAIA ◽  
V. M. MOSTEPANENKO
Keyword(s):  
Energy Density ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dimensional Space ◽  
Elliptic Functions ◽  
Dirichlet Boundary Conditions ◽  
Finite Domain ◽  
Dimensional Space Time ◽  
Elliptic Solutions ◽  
Minimal Mass

The general static solutions of the scalar field equation for the potential [Formula: see text] are determined for a finite domain in (1+1)-dimensional space–time. A family of real solutions is described in terms of Jacobi Elliptic Functions. We show that the vacuum–vacuum boundary conditions can be reached by elliptic cn-type solutions in a finite domain, such as that of the Kink, for which they are imposed at infinity. We prove uniqueness for elliptic sn-type solutions satisfying Dirichlet boundary conditions in a finite interval (box) as well the existence of a minimal mass corresponding to these solutions in a box. We defined expressions for the "topological charge," "total energy" (or classical mass) and "energy–density" for elliptic sn-type solutions in a finite domain. For large length of the box the conserved charge, classical mass and energy density of the Kink are recovered. Also, we have shown that using periodic boundary conditions the results are the same as in the case of Dirichlet boundary conditions. In the case of antiperiodic boundary conditions all elliptic sn-type solutions are allowed.

scholarly journals Explicit Solutions of the Wave Equation on Three Dimensional Space-Times: Two Examples with Dirichlet Boundary Conditions on a Disk

2006 ◽  
Vol 4 (4) ◽  
Author(s):  
Daniel Boykis ◽  
Patrick Moylan
Keyword(s):  
Wave Equation ◽  
Euclidean Space ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Flat Space ◽  
Dirichlet Boundary Conditions ◽  
Dimensional Euclidean Space ◽  
Curved Space

We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two-dimensional Euclidean space, and we also study the analogous problem on a certain curved space which is a Lorentzian variant of the 3-sphere. The curved space goes over into the usual flat space-time as the radius R of the curved space goes to infinity. We show, at least in some cases, that solutions of certain Dirichlet boundary value problems are obtained much more simply in the curved space than in the flat space. Since the flat space is the limit R → ∞ of the curved space, this gives an alternative method of obtaining solutions of a corresponding problem in Euclidean space.

scholarly journals Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media

2021 ◽  
Author(s):  
José Pinto ◽  
Rubén Aylwin ◽  
Carlos Jerez-Hanckes
Keyword(s):  
Convergence Rates ◽  
Periodic Structures ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Well Posedness ◽  
Fast Solver ◽  
Transmission Problems ◽  
Algebraic Error ◽  
Discrete Problems ◽  
Helmholtz Transmission Problems

We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cut-off frequen- cies (also known as Rayleigh-Wood frequencies), we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nystr\"om methods.

scholarly journals Blow-up Properties of a Coupled System of Reaction-Diffusion Equations

2021 ◽  
pp. 3052-3060
Author(s):  
Maan A. Rasheed
Keyword(s):  
Dirichlet Boundary ◽  
Blow Up ◽  
Dimensional Space ◽  
Single Point ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Coupled Reaction

    This paper is concerned with a Coupled Reaction-diffusion system defined in a ball with homogeneous Dirichlet boundary conditions. Firstly, we studied the blow-up set showing that, under some conditions, the blow-up in this problem occurs only at a single point. Secondly, under some restricted assumptions on the reaction terms, we established the upper (lower) blow-up rate estimates. Finally, we considered the Ignition system in general dimensional space as an application to our results.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

Sign in / Sign up

Export Citation Format

Share Document