Search

In this work we will present a method of virtual elements to approximate the solution of a polyharmonic problem (− Δ) n u = g. We will consider m + 1 auxiliary unknowns when n = 2 m + 1 , and m auxiliary unknowns for n = 2 m. In the first case (n = 2 m + 1), we will solve m fourth order problems and a second order one. In the even case, only m fourth-order problems have to be solved. Virtual element conforming discretizations are written for each fourth-order problem in C 1 , and a C 0 -virtual element method is established for the second-order problem. We also carry out the error analysis for both cases. Finally, we report a series of numerical tests to verify the performance of numerical scheme. [ABSTRACT FROM AUTHOR]

A C 0 nonconforming virtual element method (VEM) is employed for the discretization of nonselfadjoint fourth-order eigenvalue problems derived from the transmission eigenvalue problem on general polygonal meshes. Further, we linearize the model problem by introducing an auxiliary variable and thus the variational problem becomes double in size. We discretize the modified scheme by using suitable projection operators and acquire the convergence analysis under standard assumptions on the polygonal meshes. We establish that the resulting discretization provides a correct approximation of the spectrum and conclude the convergence of the eigenfunctions and eigenvalues. A variety of representative examples are examined to justify the theoretical estimates. [ABSTRACT FROM AUTHOR]

In this work, we have designed conforming and nonconforming virtual element methods (VEM) to approximate non-stationary nonlocal biharmonic equation on general shaped domain. By employing Faedo–Galerkin technique, we have proved the existence and uniqueness of the continuous weak formulation. Upon applying Brouwer's fixed point theorem, the well-posedness of the fully discrete scheme is derived. Further, following [J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math. 386 (2021) 113229], we have introduced Enrichment operator and derived a priori error estimates for fully discrete schemes on polygonal domains, not necessarily convex. The proposed error estimates are justified with some benchmark examples. [ABSTRACT FROM AUTHOR]

In this present paper, we propose and analyze a C 1 -conforming virtual element method to solve the so-called one-layer stationary quasi-geostrophic equations (QGE) with applications in the large scale wind-driven ocean circulation, formulated in terms of the stream-function. This problem corresponds to a nonlinear fourth order partial differential equation. The C 1 virtual space and the discrete scheme are built in a straightforward way due to the flexibility of the virtual approach. Under the assumption of small data, we prove well-posedness of the discrete problem by using a fixed-point strategy and under standard assumptions on the computational domain, we establish error estimates in H 2 -norm for the stream-function. Finally, we report four numerical experiments that illustrate the behavior of the proposed scheme and confirm our theoretical results on different families of polygonal meshes. [ABSTRACT FROM AUTHOR]

The transmission eigenvalue problem is a challenging model in the inverse scattering theory and has important applications in this topic. The aim of this paper is to analyze a C¹ virtual element method on polytopal meshes in Rd (d = 2, 3) for solving a quadratic and non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained by using the approximation theory for compact non-self-adjoint operators. Finally, a set of numerical tests illustrating the good performance of the virtual scheme are presented. [ABSTRACT FROM AUTHOR]

In two and three dimension we analyze discontinuous Galerkin methods (DG) for the acoustic vibration problem. Through all our study we consider an inviscid fluid, leading to a linear eigenvalue problem. The acoustic problem is written, in first place, in terms of the displacement. Under the approach of the non-compact operators theory, we prove convergence and error estimates for the method when the displacement formulation is considered. We analyze the influence of the stabilization parameter on the computation of the spectrum, where spurious eigenmodes arise when this parameter is not correctly chosen. Alternatively we present the formulation depending only on the pressure, comparing the performance of the DG methods with the pure displacement formulation. Computationally, we study the influence of the stabilization parameter on the arising of spurious eigenvalues when the spectrum is computed. Also, we report tests in two and three dimensions where convergence rates are reported, together with a comparison between the displacement and pressure formulations for the proposed DG methods. [ABSTRACT FROM AUTHOR]

We present a virtual element method (VEM) for a nonlocal reaction–diffusion system of the cardiac electric field. For this system, we analyze an |$H^1$| -conforming discretization by means of VEM that can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general |$L^p$| compactness criterion. Moreover, we obtain optimal order space-time error estimates in the |$L^2$| norm. Finally, we report some numerical tests supporting the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we introduce and analyze symmetric and nonsymmetric discontinuous Galerkin (DG) methods for the Stokes eigenvalue problem. The formulation is obtained by introducing the so-called pseudostress tensor, and thanks to the structure of the system, the velocity and pressure variables are eliminated. We propose different DG discretizations to solve the resulting spectral problem and the convergence analysis is based on the abstract spectral theory for noncompact operators. We show that the proposed method is spurious modes free and asymptotic estimates for the eigenvalues and eigenfunctions are proved if the so-called stabilization parameter is sufficiently large and the meshsize is small enough. We report some numerical experiments to assess the performance of the methods. [ABSTRACT FROM AUTHOR]

In this paper we analyze a finite element method for solving a quadratic eigenvalue problem derived from the acoustic vibration problem for a heterogeneous dissipative fluid. The problem is shown to be equivalent to the spectral problem for a non-compact operator and a thorough spectral characterization is given. The numerical discretization of the problem is based on Raviart-Thomas finite elements. The method is proved to be free of spurious modes and to converge with optimal order. Finally, we report numerical tests which allow us to assess the performance of the method. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a Virtual Element Method (VEM) for solving a non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a C 1 -conforming discretization by means of the VEM. We use the classical approximation theory for compact non-self-adjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose an H2(Ω) conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented. [ABSTRACT FROM AUTHOR]

We aim to provide a finite element analysis for the elastoacoustic vibration problem. We use a dual-mixed variational formulation for the elasticity system and combine the lowest order Lagrange finite element in the fluid domain with the reduced symmetry element known as PEERS and introduced for linear elasticity in [1]. We show that the resulting global nonconforming scheme provides a correct spectral approximation and we prove quasi-optimal error estimates. Finally, we confirm the asymptotic rates of convergence by numerical experiments. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a finite element method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. We introduce an auxiliary variable w : = Δ u (with u representing the displacement of the plate) to write a variational formulation of the spectral problem. We propose a conforming discretization of the problem, where the unknowns are approximated by piecewise linear and continuous finite elements. We show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments supporting our theoretical results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a virtual element method for the two-dimensional Steklov eigenvalue problem. We propose a discretization by means of the virtual elements presented in [L. Beirão da Veiga et al., Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199-214]. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates for the eigenfunctions and a double order for the eigenvalues. We also prove higher-order error estimates for the computation of the eigensolutions on the boundary, which in some Steklov problems (computing sloshing modes, for instance) provides the quantity of main interest (the free surface of the liquid). Finally, we report some numerical tests supporting the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of an augmented finite element approximation of the axisymmetric Brinkman equations. Stabilization of the variational formulation is achieved by adding suitable Galerkin least-squares terms, allowing us to transform the original problem into a formulation better suited for performing its stability analysis. The sought quantities (here velocity, vorticity, and pressure) are approximated by Raviart.Thomas elements of arbitrary order k≥0, piecewise continuous polynomials of degree k + 1, and piecewise polynomials of degree k, respectively. The wellposedness of the resulting continuous and discrete variational problems is rigorously derived by virtue of the classical Babu.ska.Brezzi theory. We further establish a priori error estimates in the natural norms, and we provide a few numerical tests illustrating the behavior of the proposed augmented scheme and confirming our theoretical findings regarding optimal convergence of the approximate solutions. [ABSTRACT FROM AUTHOR]

In this paper we analyse a finite element approximation of the Stokes eigenvalue problem. We introduce a variational formulation relying only on the pseudostress tensor and propose a discretization by means of the lowest-order Brezzi-Douglas-Marini mixed finite element. However, similar results hold true for other H(div)-conforming elements, like Raviart-Thomas elements. We show that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates. Finally, we report some numerical tests supporting our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a virtual element method (VEM) of high order to solve the fourth order plate buckling eigenvalue problem on polygonal meshes. We write a variational formulation based on the Kirchhoff–Love model depending on the transverse displacement of the plate. We propose a C 1 conforming virtual element discretization of arbitrary order k ≥ 2 and we use the so-called Babuška–Osborn abstract spectral approximation theory to show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the buckling modes (eigenfunctions) and a double order for the buckling coefficients (eigenvalues). Finally, we report some numerical experiments illustrating the behavior of the proposed scheme and confirming our theoretical results on different families of meshes. [ABSTRACT FROM AUTHOR]