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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

Here we discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by, we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use C 0 and C 1 virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.

42 ENGINEERING↗
DOE OSTI2018

The virtual element method for eigenvalue problems with potential terms on polytopic meshes

In this work, we extend the conforming virtual element method (VEM) to the numerical resolution of eigenvalue problems with potential terms on a polytopic mesh. An important application is that of the Schrödinger equation with a pseudopotential term. This model is a fundamental element in the numerical resolution of more complex problems from the Density Functional Theory. The VEM is based on the construction of the discrete bilinear forms of the variational formulation through certain polynomial projection operators that are directly computable from the degrees of freedom. The method shows a great flexibility with respect to the meshes and provides a correct spectral approximation with optimal convergence rates. This point is discussed from both the theoretical and the numerical viewpoint. The performance of the method is numerically investigated by solving the quantum harmonic oscillator problem with the harmonic potential and a singular eigenvalue problem with zero potential for the first eigenvalues.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A Virtual Element Method for the Full MHD system

In this manuscript we present a novel discretization for the incompressible MHD system. Our approach follows the framework of the Virtual Element Method and offers two main advantages. The method can be implemented in unstructured meshes making it highly versatile and capable of handling a wide array of problems involving interfaces, free-boundaries or adaptive refinements on the mesh. The second advantage involves the divergence of the magnetic field, our approach guarantees that it remains solenoidal. We include a theoretical proof of the condition on the magnetic field as well as energy estimates and a well-posedness study. The latter sheds light as to the stability properties of the method.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2018

Design and convergence analysis of the conforming virtual element method for polyharmonic problems

In the recent years, there has been a tremendous interest to numerical methods that approximate partial differential equations (PDEs) on computational meshes with arbitrarily-shaped polygonal/polyhedral (polytopal, for short) elements. A nonexhaustive list of such methods include the Mimetic Finite Difference method [4, 9, 21, 24, 28, 33, 35, 62, 64–67], the Polygonal Finite Element Method [68, 73, 74], the polygonal Discontinuous Galerkin Finite Element Methods [5, 7, 8, 10, 13–15, 18, 36, 38, 39, 41, 50], the Hybridizable Discontinuous Galerkin and Hybrid High-Order Methods [46–48, 51], the Gradient Discretization method [49, 53, 54], the Finite Volume Method [52], and the BEM-based FEM [71]. An alternative approach that also proved to be very successful is provided by the Virtual Element method (VEM), which was originally proposed in [19] for the numerical treatment of second-order elliptic problems [43, 44], and readily extended to linear and nonlinear elasticity [22, 25, 56], plate bending problems [37], Cahn-Hilliard equation [3], Stokes equations [2], Laplace-Beltrami equation [55], Darcy-Brinkam equation [76], discrete topology optimization problems [6], fracture networks problems [29], eigenvalue problems [59, 75]. The mixed virtual element formulation was proposed in [20, 34]; the nonconforming Virtual element formulations was proposed for second-order elliptic problems in [16], and later extended to general advection-reaction-diffusion problems, Stokes equation, the biharmonic problems, the iegenvalue problem, and the Schrodinger equation in [11, 30, 42, 45, 58, 80]. The p- and hp-version of the VEM were developed in [26, 27] and efficient multigrid methods for the resulting linear system of equations in [12]. A posteriori error estimates can be found in [40].

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Arbitrary Order Virtual Element Methods for High‐Order Phase‐Field Modeling of Dynamic Fracture

ABSTRACT Accurate modeling of fracture nucleation and propagation in brittle and ductile materials subjected to dynamic loading is important in predicting material damage and failure under extreme conditions. Phase‐field fracture models have garnered a lot of attention in recent years due to their success in representing damage and fracture processes in a wide class of materials and under a variety of loading conditions. Second‐order phase‐field fracture models are by far the most popular among researchers (and increasingly, among practitioners), but fourth‐order models have started to gain broader acceptance since their more recent introduction. The exact solution corresponding to these high‐order phase‐field fracture models has higher regularity. Thus, numerical solutions of the model equations can achieve improved accuracy and higher spatial convergence rates. In this work, we develop a virtual element framework for the high‐order phase‐field model of dynamic fracture. The virtual element method (VEM) can be regarded as a generalization of the classical finite element method. In addition to many other desirable characteristics, the VEM allows computing on polytopal meshes. Here, we use ‐conforming virtual elements and the generalized‐ time integration method for the momentum balance equation, and adopt ‐conforming virtual elements for the high‐order phase‐field equation. We verify our virtual element framework using classical quasi‐static benchmark problems and demonstrate its capabilities with the aid of numerical simulations of dynamic fracture in brittle materials.

42 ENGINEERING↗
DOE OSTI2021

A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. Finally, we numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the $\mathbb{P}_1$ – $\mathbb{P}_0$ Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method (FEM) for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular and square meshes where the method coincides with the P 1 - P 0 Scott-Vogelius scheme, which is well-known to be unstable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Design, analysis and numerical experiments for the virtual element p and hp approximations of elliptic eigenvalue problems

We discuss the p- and the hp-versions of the virtual element method for the approximation of the eigenvalues of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schr¨odinger equation with a pseudo-potential term. We present in details the analysis of the p-version of the method, proving the exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing hp-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the hp-spaces.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes

Here, in this work we develop arbitrary-order Discontinuous Skeletal Gradient Discretisations (DSGD) on general polytopal meshes. Discontinuous Skeletal refers to the fact that the globally coupled unknowns are broken polynomials on the mesh skeleton. The key ingredient is a high-order gradient reconstruction composed of two terms: (i) a consistent contribution obtained mimicking an integration by parts formula inside each element and (ii) a stabilising term for which sufficient design conditions are provided. An example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high-order residuals on a Raviart–Thomas–Nédélec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limit-conformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Lastly, links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming Virtual Element methods are also studied. Numerical examples complete the exposition.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes

We extend the Virtual Element Method to a two-dimensional unsteady nonlinear convection-diffusion equation characterized by a fractional-order derivative with respect to the time variable. Our methodology is based on three fundamental technical components: a fractional version of the Grunwald-Letnikov approximation, discrete maximal regularity, and the regularity theory associated with non-linearity. We prove the method's well-posedness, i.e., the approximate solution's existence and uniqueness to the time-fractional convection-diffusion equation with a Lipschitz nonlinear source term. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L 2 -norm and H 1 -norm to various mesh configurations is validated by numerical results, underlining the practical effectiveness of the proposed method.

97 MATHEMATICS AND COMPUTING↗