A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed ...by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete H^1 and L^2 norms are established for the corresponding weak Galerkin mixed finite element solutions.
This paper introduces a finite element method by using a weakly defined gradient operator over generalized functions. The use of weak gradients and their approximations results in a new concept ...called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations. This article intends to provide a general framework for managing differential operators on generalized functions. As a demonstrative example, the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem, in which the classical gradient operator is replaced by the discrete weak gradient. The resulting numerical scheme is called a weak Galerkin (WG) finite element method. It can be seen that the weak Galerkin method allows the use of totally discontinuous functions in the finite element procedure. For the second order elliptic problem, an optimal order error estimate in both a discrete H1 and L2 norms are established for the corresponding weak Galerkin finite element solutions. A superconvergence is also observed for the weak Galerkin approximation.
MFEM: A modular finite element methods library Anderson, Robert; Andrej, Julian; Barker, Andrew ...
Computers & mathematics with applications (1987),
01/2021, Letnik:
81, Številka:
1
Journal Article
Recenzirano
Odprti dostop
MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide ...variety of discretization approaches and emphasis on usability, portability, and high-performance computing efficiency. MFEM’s goal is to provide application scientists with access to cutting-edge algorithms for high-order finite element meshing, discretizations and linear solvers, while enabling researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel and GPU-accelerated settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library.
Abstract
In the context of unfitted finite element discretizations, the realization of high-order methods is challenging due to the fact that the geometry approximation has to be sufficiently ...accurate. We consider a new unfitted finite element method that achieves a high-order approximation of the geometry for domains that are implicitly described by smooth-level set functions. The method is based on a parametric mapping, which transforms a piecewise planar interface reconstruction to a high-order approximation. Both components, the piecewise planar interface reconstruction and the parametric mapping, are easy to implement. In this article, we present an a priori error analysis of the method applied to an interface problem. The analysis reveals optimal order error bounds for the geometry approximation and for the finite element approximation, for arbitrary high-order discretization. The theoretical results are confirmed in numerical experiments.
A preasymptotic error analysis of the finite element method (FEM) and some continuous interior penalty finite element method (CIP-FEM) for the Helmholtz equation in two and three dimensions is ...proposed. H1- and L2-error estimates with explicit dependence on the wave number k are derived. In particular, it is shown that if k2p+1 h2p is sufficiently small, then the pollution errors of both methods in H1-norm are bounded by O(k2p+1 h2p), which coincides with the phase error of the FEM obtained by existent dispersion analyses on Cartesian grids, where h is the mesh size, and p is the order of the approximation space and is fixed. The CIP-FEM extends the classical one by adding more penalty terms on jumps of higher (up to pth order) normal derivatives in order to reduce efficiently the pollution errors of higher order methods. Numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the CIP-FEM in reducing the pollution effect.
In this paper we introduce the $hp$-version discontinuous Galerkin composite finite element method for the discretization of second-order elliptic partial differential equations. This class of ...methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. The key idea in the construction of this latter class of methods is that the computational domain $\Omega$ is no longer resolved by the mesh; instead, the finite element basis (or shape) functions are adapted to the geometric details present in $\Omega$. In this paper, we extend these ideas to the discontinuous Galerkin setting, based on employing the $hp$-version of the finite element method. Numerical experiments highlighting the practical application of the proposed numerical scheme will be presented. PUBLICATION ABSTRACT
We consider adaptive least-squares finite element methods. First, we develop a guaranteed upper bound for the dual error in the L2 norm, and this can be used as a stopping criterion for the adaptive ...procedures. Secondly, based on the a posteriori error estimates for the dual variable, we develop an error indicator that identifies the local area to refine, and establish the convergence of the adaptive procedures based on the Dörfler's marking strategy. Our convergence analysis is valid for the entire range of the bulk parameter 0<Θ≤1 and it shows the effect of bulk parameter and reduction factor of elements on the convergence rate. Confirming numerical experiments are provided.
Extension operators for trimmed spline spaces Burman, Erik; Hansbo, Peter; Larson, Mats G. ...
Computer methods in applied mechanics and engineering,
01/2023, Letnik:
403
Journal Article
Recenzirano
Odprti dostop
We develop a discrete extension operator for trimmed spline spaces consisting of piecewise polynomial functions of degree p with k continuous derivatives. The construction is based on polynomial ...extension from neighboring elements together with projection back into the spline space. We prove stability and approximation results for the extension operator. Finally, we illustrate how we can use the extension operator to construct a stable cut isogeometric method for an elliptic model problem.
Nektar++ is an open-source framework that provides a flexible, high-performance and scalable platform for the development of solvers for partial differential equations using the high-order ...spectral/hp element method. In particular, Nektar++ aims to overcome the complex implementation challenges that are often associated with high-order methods, thereby allowing them to be more readily used in a wide range of application areas. In this paper, we present the algorithmic, implementation and application developments associated with our Nektar++ version 5.0 release. We describe some of the key software and performance developments, including our strategies on parallel I/O, on in situ processing, the use of collective operations for exploiting current and emerging hardware, and interfaces to enable multi-solver coupling. Furthermore, we provide details on a newly developed Python interface that enables a more rapid introduction for new users unfamiliar with spectral/hp element methods, C++ and/or Nektar++. This release also incorporates a number of numerical method developments – in particular: the method of moving frames (MMF), which provides an additional approach for the simulation of equations on embedded curvilinear manifolds and domains; a means of handling spatially variable polynomial order; and a novel technique for quasi-3D simulations (which combine a 2D spectral element and 1D Fourier spectral method) to permit spatially-varying perturbations to the geometry in the homogeneous direction. Finally, we demonstrate the new application-level features provided in this release, namely: a facility for generating high-order curvilinear meshes called NekMesh; a novel new AcousticSolver for aeroacoustic problems; our development of a ‘thick’ strip model for the modelling of fluid–structure interaction (FSI) problems in the context of vortex-induced vibrations (VIV). We conclude by commenting on some lessons learned and by discussing some directions for future code development and expansion.
Program Title: Nektar++
Program Files doi:http://dx.doi.org/10.17632/9drxd9d8nx.1
Code Ocean Capsule:https://doi.org/10.24433/CO.9865757.v1
Licensing provisions: MIT
Programming language: C++
External routines/libraries: Boost, METIS, FFTW, MPI, Scotch, PETSc, TinyXML, HDF5, OpenCASCADE, CWIPI
Nature of problem: The Nektar++ framework is designed to enable the discretisation and solution of time-independent or time-dependent partial differential equations.
Solution method: spectral/hp element method
Problems involving material interfaces are challenging, owing to the cumbersome requirements in meshing such as a fine matching mesh on both sides of the interface. Often one does not require very ...fine meshes on either side of the interface owing to different geometric and material properties. If the interface meshes are finer on either side, the degrees of freedom (DOFs) increases substantially. To address this difficulty, in this work the benefits of polygonal elements constructed based on the cell-based smoothed finite element method (SFEM) are explored and an alternate approach for modelling fracture along cohesive interfaces is introduced. The proposed approach greatly reduces the DOFs in the system by allowing for more nodes along the required interface region, whilst allowing for a coarse mesh elsewhere in an elegant way, without compromising the accuracy compared with the conventional finite element method. The proposed framework enables the interface to be represented independent of the meshes at the interface giving complete freedom on meshing. The robustness, accuracy and the convergence properties are demonstrated with a few numerical tests involving straight, curved and multiple interfaces.
•n−sided cell-based smoothed finite element formulated for non-matching grids.•Non-matching grids considered for modelling interfacial cracks both straight, curved and multiple interfaces.•Proposed framework reduces the computational burden since transition elements are not required when polygonal elements are used.•Computation of Jacobian is not required due to the use of SFEM.
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