Elliptic Boundary Value Problems Research Articles

On Some Convergence Properties for Finite Element Approximations to the Inverse of Linear Elliptic Operators

This paper deals with convergence theorems of the Galerkin finite element approximation for the second-order elliptic boundary value problems. Under some quite general settings, we show not only the pointwise convergence but also prove that the norm of approximate operator converges to the corresponding norm for the inverse of a linear elliptic operator. Since the approximate norm estimates of linearized inverse operator play an essential role in the numerical verification method of solutions for non-linear elliptic problems, our result is also important in terms of guaranteeing its validity. Furthermore, the present method can also be applied to more general elliptic problems, e.g., biharmonic problems and so on.

Well-posedness of Tricomi–Gellerstedt–Keldysh-type fractional elliptic problems

We study Tricomi–Gellerstedt–Keldysh-type fractional elliptic equations and obtain results on the well-posedness of fractional elliptic boundary value problems for general positive operators with discrete spectrum and for Fourier multipliers with positive symbols. As examples, we discuss results in half-cylinder, star-shaped graph, half-space and other domains.

Spectral problem for the Laplacian and a selfadjoint nonlinear elliptic boundary value problem

In this paper, we present some connections between the spectral problem,
 −Δu(x) = λ1u(x) in Ω,u(x) = 0 on ∂Ω
 and selfadjoint boundary value problem,
 Δu(x) − λ1u(x) + g(x, u(x)) = h(x) in Ω,u(x) = 0 on ∂Ω,
 where λ1 is the smallest eigenvalue of −∆, Ω ⊆ Rn is a bounded domain, h ∈ L2(Ω) and the nonlinear function g is a Caratheodory function satisfying a growth condition. We initially investigate the existence of solutions for the spectral problem by considering the selfadjoint boundary value problem. The selfadjoint boundary value problem is then considered for both existence and estimation results. We use degree argument in order to show that the selfadjoint boundary value problem has a solution instead of the Landesman-Lazer condition or the monotonocity assumption on the second argument of the function g.
 In this paper, we present some connections between the spectral problem,
 and selfadjoint boundary value problem,
 where λ1 is the smallest eigenvalue of −∆, Ω ⊆ Rn is a bounded domain, h ∈ L2(Ω) and the nonlinear function g is a Caratheodory function satisfying a growth condition. We initially investigate the existence of solutions for the spectral problem by considering the selfadjoint boundary value problem. The selfadjoint boundary value problem is then considered for both existence and estimation results. We use degree argument in order to show that the selfadjoint boundary value problem has a solution instead of the Landesman-Lazer condition or the monotonocity assumption on the second argument of the function g.

A pretraining domain decomposition method using artificial neural networks to solve elliptic PDE boundary value problems

Developing methods of domain decomposition (DDM) has been widely studied in the field of numerical computation to estimate solutions of partial differential equations (PDEs). Several case studies have also reported that it is feasible to use the domain decomposition approach for the application of artificial neural networks (ANNs) to solve PDEs. In this study, we devised a pretraining scheme called smoothing with a basis reconstruction process on the structure of ANNs and then implemented the classic concept of DDM. The pretraining process that is engaged at the beginning of the training epochs can make the approximation basis become well-posed on the domain so that the quality of the estimated solution is enhanced. We report that such a well-organized pretraining scheme may affect any NN-based PDE solvers as we can speed up the approximation, improve the solution’s smoothness, and so on. Numerical experiments were performed to verify the effectiveness of the proposed DDM method on ANN for estimating solutions of PDEs. Results revealed that this method could be used as a tool for tasks in general machine learning.

Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes

This article considers the extension of two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when agglomerated polygonal/polyhedral meshes are employed for the coarse mesh approximation. We recall that within the two-grid setting, while it is necessary to solve a nonlinear problem on the coarse approximation space, only a linear problem must be computed on the original fine finite element space. In this article, the coarse space will be constructed by agglomerating elements from the original fine mesh. Here, we extend the existing a priori and a posteriori error analysis for the two-grid hp-version discontinuous Galerkin finite element method from Congreve et al. [1] for coarse meshes consisting of standard element shapes to include arbitrarily agglomerated coarse grids. Moreover, we develop an hp-adaptive two-grid algorithm to adaptively design the fine and coarse finite element spaces; we stress that this is undertaken in a fully automatic manner, and hence can be viewed as blackbox solver. Numerical experiments are presented for two- and three-dimensional problems to demonstrate the computational performance of the proposed hp-adaptive two-grid method.

Multi-Fidelity Machine Learning Applied to Steady Fluid Flows

A machine learning method to predict steady external fluid flows using elliptic input features is introduced. Using data from as few as one high-fidelity simulation, the proposed method produces models generalisable under changes to boundary geometry by using solutions to elliptic boundary value problems over the flow domain as the model input, instead of Cartesian coordinates of the domain. Training data is generated through pointwise evaluation of flow features at points selected through a quad-tree adaptive sampling method to concentrate training points in areas with large field gradients. Models are trained within a training window around the body, while predictions are smoothly extended to freestream conditions using a Partition-of-Unity extension. Predictive capabilities of the machine learning model are demonstrated in steady-state flow of incompressible fluid around a cylinder and a Joukowski airfoil. The predicted flow field is used to warm-start CFD simulations to achieve acceleration in solver convergence.

Unsaturated Algorithms for the Numerical Solution of Elliptic Boundary Value Problems in Smooth Axisymmetric Domains

Unsaturated Algorithms for the Numerical Solution of Elliptic Boundary Value Problems in Smooth Axisymmetric Domains

Variational Physics Informed Neural Networks: the Role of Quadratures and Test Functions

In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.

Infinitely many weak solutions for a p-triharmonic problem with Navier boundary conditions

ABSTRACT The existence of infinitely many weak solutions to a class of nonlinear elliptic Navier boundary value problems containing the p-triharmonic operator is considered by a variational approach and critical point theory.

Parallelized discrete exterior calculus for three-dimensional elliptic problems

A formulation of elliptic boundary value problems is used to develop the first discrete exterior calculus (DEC) library for massively parallel computations with 3D domains. This can be used for steady-state analysis of any physical process driven by the gradient of a scalar quantity, e.g. temperature, concentration, pressure or electric potential, and is easily extendable to transient analysis. In addition to offering this library to the community, we demonstrate one important benefit from the DEC formulation: effortless introduction of strong heterogeneities and discontinuities. These are typical for real materials, but challenging for widely used domain discretization schemes, such as finite elements. Specifically, we demonstrate the efficiency of the method for calculating the evolution of thermal conductivity of a solid with a growing crack population. Future development of the library will deal with transient problems, and more importantly with processes driven by gradients of vector quantities. Program summaryProgram Title: ParaGEMSCPC Library link to program files:https://doi.org/10.17632/t8rpkc7xnx.1Developer's repository link:https://bitbucket.org/pieterboom/paragems/Licensing provisions: BSD 2-clauseProgramming language: Fortran 90External routines/libraries: BLAS/LAPACK, MPI, PETSc [1], hypre ParaSails [2,3]Nature of problem: Large scale simulation of 3D elliptic boundary value problems in discrete media with heterogeneous or discontinuous properties.Solution method: ParaGEMS is a library implementing discrete exterior calculus (DEC) for distributed memory architectures. It is distributed with mini-applications for solving 3D elliptic boundary value problems, which describe a number of physical laws (Fourier's law, Fick's law, Darcy's law and Ohm's law), in discrete media with heterogeneous or discontinuous properties.Additional comments including restrictions and unusual features: ParaGEMS requires meshes generated in the default format used by Triangle [4] and TetGen [5].

A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal control

A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal control

On a degenerate singular elliptic problem

AbstractIn this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by where Ω is a bounded smooth domain in with , w belongs to the Muckenhoupt class for some , f is a nonnegative function belonging to some Lebesgue space and .

$$C^*$$-Algebras of Transmission Problems and Elliptic Boundary Value Problems with Shift Operators

$$C^*$$-Algebras of Transmission Problems and Elliptic Boundary Value Problems with Shift Operators

Three-dimensional Haar wavelet method for singularly perturbed elliptic boundary value problems on non-uniform meshes

Three-dimensional Haar wavelet method for singularly perturbed elliptic boundary value problems on non-uniform meshes

Parameter identification for elliptic boundary value problems: an abstract framework and applications

Parameter identification problems for partial differential equations are an important subclass of inverse problems. The parameter-to-state map, which maps the parameter of interest to the respective solution of the PDE or state of the system, plays the central role in the (usually nonlinear) forward operator. Consequently, one is interested in well-definedness and further analytic properties such as continuity and differentiability of this operator w.r.t. the parameter in order to make sure that techniques from inverse problems theory may be successfully applied to solve the inverse problem. In this work, we present a general functional analytic framework suited for the study of a huge class of parameter identification problems including a variety of elliptic boundary value problems with Dirichlet, Neumann, Robin or mixed boundary conditions in Hilbert and Banach spaces and possibly complex-valued parameters. In particular, we show that the corresponding parameter-to-state operators fulfill, under suitable conditions, the tangential cone condition, which is often postulated for numerical solution techniques. This framework particularly covers the inverse medium problem and an inverse problem that arises in terahertz tomography.

A suite of Richardson preconditioners for semi-implicit all-scale atmospheric models

The paper documents a suite of preconditioners for Krylov-subspace solvers of elliptic boundary-value problems (BVPs) that underlie semi-implicit integrations of the nonhydrostatic equations governing the dynamics of all-scale atmospheric flows. Effective preconditioning of the linear operators inherent in the semi-implicit models lies at the heart of the state-of-the-art multiscale-flow simulation. This is especially evident in simulations of global weather and climate—posed on a thin spherical shell—where some form of direct tridiagonal inversion of the operator in the vertical is crucial to relax the often enormous stiffness of the problem. The documented preconditioners stem from the Richardson's (1910) idea of augmenting an elliptic BVP with a transient diffusion equation. Exploiting this idea for mixed explicit-implicit pseudo-time-stepping schemes leads to a broad suite of stationary-iteration solvers, including the many classical algorithms. Here, the high-performance all-scale EULAG model (Smolarkiewicz et al. (2014) [58]), with a flexible three-dimensional decomposition of MPI tasks, is furnished with the preconditioners akin to the classical alternating-direction-implicit (ADI) algorithms, generalized to optional permutations of parallel tridiagonal inversions. The utility of various options is found to be problem dependent, in terms of computational accuracy as well as efficiency. The main thrust of the work is on the long-range forecasts using large anisotropic grids. The relative efficiency and/or accuracy gains attainable with the developed preconditioners are illustrated for idealized scenarios representative of atmospheric flows from planetary to a single-cloud and laboratory scales. The key insight that best encapsulates the significance and novelty of the present work is that there is no single “super-preconditioner” that will perform best in all cases, yet the suite as a whole offers substantial gains in the model performance.

A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems

In this article, we propose a simplified radial basis function (RBF) method with exterior fictitious sources for solving elliptic boundary value problems (BVPs). Three simplified RBFs, including Gaussian, multiquadric (MQ), and inverse multiquadric (IMQ) without the shape parameter, are adopted in this study. With the consideration of many exterior fictitious sources outside the domain, the radial distance of the RBF is always greater than zero, such that we can remove the shape parameter from RBFs. Additionally, simplified Gaussian, MQ, and IMQ RBFs and their derivatives in the governing equation are always smooth and nonsingular. Comparative analysis is conducted for three different collocation types, including conventional uniform centers, randomly fictitious centers, and exterior fictitious sources. Numerical examples of elliptic BVPs in two and three dimensions are carried out. The results demonstrate that the proposed simplified RBFs with exterior fictitious sources can significantly improve the accuracy, especially for the Laplace equation. Furthermore, the proposed simplified RBFs exhibit the simplicity of solving elliptic BVPs without finding the optimum shape parameter.

Sobolev gradient type iterative solution methods for a nonlinear 4th order elastic plate equation

This paper considers the numerical solution of an elastic bending model of a thin plate, based on material nonlinearity, which leads to a nonlinear elliptic 4th order boundary value problem. Our goal is to summarize possible efficient iterative solvers, adapted to this problem, based on a Sobolev space background. It is proved that the proposed methods exhibit robust behaviour, that is, convergence rates are bounded independently of the considered Galerkin discretization subspace.

On the strong comparison principle for degenerate elliptic problems with convection

The weak and strong comparison principles (WCP and SCP, respectively) are investigated for quasilinear elliptic boundary value problems with the p-Laplacian in one space dimension, Δp(u)=defddx(|u′|p−2u′). We treat the “degenerate” case of 2<p<∞ and allow also for the nontrivial convection velocityb:[−1,1]→R in the underlying domain Ω=(−1,1). We establish the WCP under a rather general, “natural sufficient condition” on the convection velocity, b(x), and the reaction function, φ(x,u). Furthermore, we establish also the SCP under a number of various additional hypotheses. In contrast, with these hypotheses being violated, we construct also a few rather natural counterexamples to the SCP and discuss their applications to an interesting classical problem of fluid flow in porous medium, “seepage flow of fluids in inclined bed”. Our methods are based on a mixture of classical and new techniques.

A two‐weight boundedness criterion on spaces of homogeneous type with its application to some elliptic boundary value problems

Let be a space of homogeneous type in the sense of Coifman and Weiss. Namely, is a non‐empty set, is a quasi‐metric on , and is a positive measure satisfying the doubling condition. In this article, the authors establish a general weighted good‐ inequality for a pair of functions on . Using this weighted good‐ inequality, the authors further obtain a two‐weight boundedness criterion for a pair of functions on in the scale of weighted Lebesgue spaces. As an application, the authors prove the two‐weight global gradient estimates for some quasi‐linear elliptic equations on bounded Reifenberg flat domains of .