Order Elliptic Boundary Value Problems Research Articles

Approximate controllability of a semilinear elliptic problem in periodically perforated domains

In this paper, we study the controllability of a semilinear second order elliptic boundary value problem in a periodically perforated domain, where the control is distributed in an open subset of the domain. Compared to existing literature, the holes in this control problem can intersect the control zone. We study L2 and H1-approximate controllability for the problem and we prove the homogenization results for the control problem in an ε-periodically perforated domain. The holes are of same size as ε with the coefficients of state equation, and the cost functional being rapidly oscillating.

A meshfree method for the solution of 2D and 3D second order elliptic boundary value problems in heterogeneous media

We propose a simple meshfree method for two-dimensional and three-dimensional second order elliptic boundary value problems in heterogeneous media based on equilibrated basis functions. The domain is discretized by a regular nodal grid, over which the degrees of freedom are defined. The boundary is also discretized by simply introducing some boundary points over it, independent from the domain nodes, granting the method the ability of application for arbitrarily shaped domains without the drawback of irregularity in the nodal grid. In heterogeneous media, the governing Partial Differential Equation (PDE) has non-constant coefficients for the partial derivatives, preventing Trefftz-based techniques to be applicable. The proposed method satisfies the PDE independent from the boundary conditions, as in Trefftz approaches. Meanwhile, by applying the weak form of the PDE instead of the strong form through weighted residual integration, the inconvenience of variable coefficients shall be resolved, since the bases will not need to analytically satisfy the PDE. The weighting in the weak form integrals is such that the boundary integrals vanish. The boundary conditions are thus collocated over the defined boundary points. Domain integrals break into algebraic combination of one-dimensional pre-evaluated integrals, thus omitting the numerical integration from the solution process. Each node corresponds to a local sub-domain called cloud. The overlap between adjacent clouds ensures integrity of the solution and its derivatives throughout the domain, an advantage with respect to C0 formulations.

Regular solutions to elliptic equations

A review of results and techniques on the existence of regular radial solutions to second order elliptic boundary value problems driven by linear and quasilinear operators is presented. Of particular interest are results where the solvability of a given elliptic problem can be analyzed by the relationship between the spectrum of the operator and the behavior of the nonlinearity near infinity and at zero. Energy arguments and Pohozaev type identities are used extensively in that analysis. An appendix with a proof of the contraction mapping principle best suited for using continuous dependence to ordinary differential equations on initial conditions is presented. Another appendix on the phase plane analysis as needed to take advantage of initial conditions is also included. For studies on singular solutions the reader is referred to Ardila et al., Milan J. Math (2014) and references therein. See also https://ejde.math.txstate.edu/special/02/c2/abstr.html

Realisations of elliptic operators on compact manifolds with boundary

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of Bär-Ballmann to first order elliptic operators. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calderón projectors which, in the first order case, is equivalent to results of Bär-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven.

Training RBF neural networks for the solution of elliptic boundary value problems

We propose a radial basis function (RBF) neural network method for solving two- and three–dimensional second and fourth order elliptic boundary value problems (BVPs). The neural network in question is trained by minimizing a nonlinear least squares functional, thus determining the optimal values of the various RBF parameters involved. The functional minimization is carried out using standard MATLAB® software efficiently. Several numerical experiments are presented to demonstrate the efficacy of the proposed method.

A C0 Interior Penalty Discontinuous Galerkin Method and an equilibrated a posteriori error estimator for a nonlinear fourth order elliptic boundary value problem of p-biharmonic type

We consider a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of a nonlinear fourth order elliptic boundary value problem of p-biharmonic type and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a discretization of the corresponding minimization problem involving a suitably defined reconstruction operator. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W2,p norm in terms of the associated primal and dual energy functionals. It requires the construction of an equilibrated flux and an equilibrated moment tensor based on a three-field formulation of the C0IPDG approximation. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.

An isogeometric mortar method for the coupling of multiple NURBS domains with optimal convergence rates

We investigate the mortar finite element method for second order elliptic boundary value problems on domains which are decomposed into patches Omega _k with tensor-product NURBS parameterizations. We follow the methodology of IsoGeometric Analysis (IGA) and choose discrete spaces X_{h,k} on each patch Omega _k as tensor-product NURBS spaces of the same or higher degree as given by the parameterization. Our work is an extension of Brivadis et al. (Comput Methods Appl Mech Eng 284:292–319, 2015) and highlights several aspects which did not receive full attention before. In particular, by choosing appropriate spaces of polynomial splines as Lagrange multipliers, we obtain a uniform infsup-inequality. Moreover, we provide a new additional condition on the discrete spaces X_{h,k} which is required for obtaining optimal convergence rates of the mortar method. Our numerical examples demonstrate that the optimal rate is lost if this condition is neglected.

An Inverse Problem Solution for Thermal Conductivity Reconstruction

This work deals with the inverse problem of reconstructing the thermal conductivity coefficient of the (2+1)D heat equation from over–posed data at the boundaries. The proposed solution uses a variational approach for identifying the coefficient. The inverse problem is reformulated as a higher–order elliptic boundary–value problem for minimization of a quadratic functional of the original equation. The resulting system consists of a well–posed fourth–order boundary–value problem for the temperature and an explicit equation for the unknown thermal conductivity coefficient. The existence and uniqueness of the resulting higher–order boundary–value problem are investigated. The unique solvability of the inverse coefficient problem is proven. The numerical algorithm is validated and applied to problems of reconstructing continuous nonlinear coefficient and discontinuous coefficients. Accurate and stable numerical solutions are obtained.

A Survey on Numerical Methods for Spectral Space-Fractional Diffusion Problems

AbstractThe survey is devoted to numerical solution of the equation $ {\mathcal A}^\alpha u=f $, 0 < α<1, where $ {\mathcal A} $ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝd. The fractional power $ {\mathcal A}^\alpha $ is a non-local operator and is defined though the spectrum of $ {\mathcal A} $. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator $ {\mathcal A} $ by using an N-dimensional finite element space Vh or finite differences over a uniform mesh with N points. In the case of finite element approximation we get a symmetric and positive definite operator $ {\mathcal A}_h: V_h \to V_h $, which results in an operator equation $ {\mathcal A}_h^{\alpha} u_h = f_h $ for uh ∈ Vh.The numerical solution of this equation is based on the following three equivalent representations of the solution: (1) Dunford-Taylor integral formula (or its equivalent Balakrishnan formula, (2.5), (2) extension of the a second order elliptic problem in Ω × (0, ∞)⊂ ℝd+1 [17,55] (with a local operator) or as a pseudo-parabolic equation in the cylinder (x, t) ∈ Ω × (0, 1), [70, 29], (3) spectral representation (2.6) and the best uniform rational approximation (BURA) of zα on [0, 1], [37,40].Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of $ {\mathcal A}_h^{-\alpha} $. In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.

Stabilizer-free weak Galerkin methods for monotone quasilinear elliptic PDEs

In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain assumptions on the nonlinear coefficient, we show that the discrete problem has a unique solution. This is achieved by showing that the associated operator satisfies certain continuity and monotonicity properties. With the help of these properties, we derive optimal error estimates in the energy norm. We present several numerical examples to verify the error estimates.

Wavelet kernel function based multiscale LSSVM for elliptic boundary value problems

Recently, Mehrkanoon and Suykens designed a least square support vector machine (LSSVM) for learning solutions to partial differential equations in [1], where the Gaussian radial basis function is used as the kernel of the LSSVM. The purpose of the present paper is twofold: firstly, we extend the Gaussian kernel to wavelet kernel; and secondly, we propose a multiscale scheme, by noticing the multiscale nature of the wavelet kernel functions. The multiscale algorithm consists of a sequence of residual corrections to the solution of the partial differential equation, in which different scale parameters are employed to accommodate information at different scales. However, the direct computation of the multiscale problem would be difficult due to the mixing of different scales. To resolve this problem, we further introduce a multilevel algorithm, which decompose the multiscale algorithm into multiple levels: on the first level, a coarse data set and a large scale parameter are chosen and the target function is interpolated in this data set to capture the large-scale variations of the target function; next, on the second level, a smaller scale parameter is used to interpolate the residuals on a finer data set, capturing the finer details. The sum of both interpolants obviously better approximates the target function at the data sites on the finer data set. This process can be further applied to finer and finer scales till the anticipated accuracy is achieved. Note that sometimes the correction on the finer scale is necessary only at some local sites, thus an adaptive algorithm with only local corrections on finer scale is introduced. The numerical tests on some linear second order elliptic boundary value problems show the efficiency of the multilevel algorithm and the adaptive algorithm.

A Petrov–Galerkin multilayer discretization to second order elliptic boundary value problems

We study in this paper a multilayer discretization of second order elliptic problems, aimed at providing reliable multilayer discretizations of shallow fluid flow problems with diffusive effects. This discretization is based upon the formulation by transposition of the equations. It is a Petrov–Galerkin discretization in which the trial functions are piecewise constant per horizontal layers, while the test functions are continuous piecewise linear, on a vertically shifted grid.We prove the well posedness and optimal error order estimates for this discretization in natural norms, based upon specific inf–sup conditions.We present some numerical tests with parallel computing of the solution based upon the multilayer structure of the discretization, for academic problems with smooth solutions, with results in full agreement with the theory developed.

Systems of elliptic boundary value problems and applications to competition models

We generalize the main result on existence of nonzero nonnegative solutions of systems of second order elliptic boundary value problems obtained by Lan (2011). The motivation for the generalization is to propose and study the competition models of Ricker and Beverton–Holt types governed by such systems. To the best of our knowledge, there is little study on such continuous competition models although such difference equation and first order ordinary differential equation competition models have been widely studied. Our results enrich and develop the connections among the classical fixed point index theory, systems of second order elliptic boundary value problems and population dynamics.

Diagonalized Legendre spectral methods using Sobolev orthogonal polynomials for elliptic boundary value problems

Fully diagonalized spectral methods using Sobolev orthogonal/biorthogonal basis functions are proposed for solving second order elliptic boundary value problems. We first construct the Fourier-like Sobolev polynomials which are mutually orthogonal (resp. bi-orthogonal) with respect to the bilinear form of the symmetric (resp. unsymmetric) elliptic Neumann boundary value problems. The exact and approximation solutions are then expanded in an infinite and truncated series in the Sobolev orthogonal polynomials, respectively. An identity is also established for the a posterior error estimate with a simple error indicator. Further, the Fourier-like Sobolev orthogonal polynomials and the corresponding Legendre spectral method are proposed in parallel for Dirichlet boundary value problems. Numerical experiments illustrate that our Legendre methods proposed are not only efficient for solving elliptic problems but also equally applicable to indefinite Helmholtz equations and singular perturbation problems.

Elliptic boundary value problems with Gaussian white noise loads

Linear second order elliptic boundary value problems (BVP) on bounded Lipschitz domains are studied in the case of Gaussian white noise loads. The challenging cases of Neumann and Robin BVPs are considered.The main obstacle for usual variational methods is the irregularity of the load. In particular, the Neumann boundary values are not well-defined.In this work, the BVP is formulated by replacing the continuity of boundary trace mappings with measurability. Instead of variational methods alone, the novel BVP derives also from Cameron–Martin space techniques.The new BVP returns the study of irregular white noise to the study of L2-loads.

Incomplete factorization by local exact factorization (ILUE)

This study proposes a new preconditioning strategy for symmetric positive (semi-)definite SP(S)D matrices referred to as incomplete factorization by local exact factorization (ILUE). The investigated technique is based on exact LU decomposition of small-sized local matrices associated with a splitting of the domain into overlapping or non-overlapping subdomains. The ILUE preconditioner is defined and its relative condition number estimated. Numerical tests on linear systems arising from the finite element (FE) discretization of a second order elliptic boundary value problem in mixed form demonstrate the advantage of the new algorithm, even for problems with highly oscillatory permeability coefficients, against the classical ILU(p) and ILUT(τ) incomplete factorization preconditioners.

Approximation of a Second-order Elliptic Equation with Discontinuous and Highly Oscillating Coefficients by Finite Volume Methods

In this paper we consider the numerical approximation of a class of second order elliptic boundary value problems with discontinuous and highly periodically oscillating coefficients. We apply both classical and modified finite volume methods for the approximate solution of this problem. Error estimates depending on $\varepsilon$ the parameter involved in the periodic homogenization are established. Numerical simulations for one-dimensional problem confirm the theorical results and also show that the modified scheme has a smaller constant of convergence than the classical scheme based on harmonic averaging for this class of equations.

The local ultraconvergence for high-degree Galerkin finite element methods

In this paper, we investigate the local ultraconvergence of k-degree (k≥3) finite element methods for the second order elliptic boundary value problem with constant coefficients over a family of uniform rectangular/triangular meshes Th on a bounded rectangular domain D. The k-degree finite element estimates are developed for the Green's function and its derivatives. They are employed to explore the relationship among dist(x,∂D), dist(x,M) and the ultraconvergence of k-degree finite element methods at vertex x, where M is the set of corners of D. Numerical examples are conducted to demonstrate our theoretical results.

On the magnitudes of compact sets in Euclidean spaces

The notion of the magnitude of a metric space was introduced by Leinster and developed in works by Leinster, Meckes and Willerton, but the magnitudes of familiar sets in Euclidean space are only understood in relatively few cases. In this paper we study the magnitudes of compact sets in Euclidean spaces. We first describe the asymptotics of the magnitude of such sets in both the small- and large-scale regimes. We then consider the magnitudes of compact convex sets with nonempty interior in Euclidean spaces of odd dimension, and relate them to the boundary behaviour of solutions to certain naturally associated higher order elliptic boundary value problems in exterior domains. We carry out calculations leading to an algorithm for explicit evaluation of the magnitudes of balls, and this establishes the convex magnitude conjecture of Leinster and Willerton in the special case of balls in dimension three. In general the magnitude of an odd-dimensional ball is a rational function of its radius, thus disproving the general form of the Leinster-Willerton conjecture. In addition to Fourier-analytic and PDE techniques, the arguments also involve some combinatorial considerations.

Analysis of the Diffuse Domain Method for Second Order Elliptic Boundary Value Problems

The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper, we study the diffuse domain method for approximating second order elliptic boundary value problems posed on bounded domains and show convergence and rates of the approximations generated by the diffuse domain method to the solution of the original second order problem when complemented by Robin, Dirichlet or Neumann conditions. The main idea of the diffuse domain method is to relax these boundary conditions by introducing a family of phase-field functions such that the variational integrals of the original problem are replaced by a weighted average of integrals of perturbed domains. From a functional analytic point of view, the phase-field functions naturally lead to weighted Sobolev spaces for which we present trace and embedding results as well as various types of Poincare inequalities with constants independent of the domain perturbations. Our convergence analysis is carried out in such spaces as well, but allows to draw conclusions also about unweighted norms applied to restrictions on the original domain. Our convergence results are supported by numerical examples.