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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 H1H^1 and L2L^2 norms are established for the corresponding weak Galerkin mixed finite element solutions.

We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.

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]

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.

We present the nonconforming virtual element method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable nonpolynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the nonpolynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two- and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the nonconforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.

The Finite Element Method (FEM) has become an indispensable technology for the modelling and simulation of engineering systems.Written for engineers and students alike, the aim of the book is to provide the necessary theories and techniques of the FEM for readers to be able to use a commercial FEM package to solve primarily linear problems in.

In the finite element method (FEM), a necessary condition for a four-node isoparametric element is that no interior angle is greater than 180° and the positivity of Jacobian determinant should be ensured in numerical implementation. In this paper, we incorporate cell-wise strain smoothing operations into conventional finite elements and propose the smoothed finite element method (SFEM) for 2D elastic problems. It is found that a quadrilateral element divided into four smoothing cells can avoid spurious modes and gives stable results for integration over the element. Compared with original FEM, the SFEM achieves more accurate results and generally higher convergence rate in energy without increasing computational cost. More importantly, as no mapping or coordinate transformation is involved in the SFEM, its element is allowed to be of arbitrary shape. Hence the restriction on the shape bilinear isoparametric elements can be removed and problem domain can be discretized in more flexible ways, as demonstrated in the example problems.

Mixed finite element methods with flux errors in H(div)-norms and div-least-squares finite element methods require a separate marking strategy in obligatory adaptive mesh-refining. The refinement indicator σ²(T,K) = η²(T, K) + µ²(K) of a finite element domain K in an admissible triangulation T consists of some residual-based error estimator η(T,K) with some reduction property under local mesh-refining and some data approximation error µ(K). Separate marking means either Dörfler marking if µ²(T) ≤ κη²(T) or otherwise an optimal data approximation algorithm with controlled accuracy. The axioms are sufficient conditions on the estimators η(T, K) and data approximation errors µ(K) for optimal asymptotic convergence rates. The enfolded set of axioms of this paper simplifies [C. Carstensen, M. Feischl, M. Page, and D. Praetorius, Comput. Math. Appl., 67 (2014), pp. 1195-1253] for collective marking, treats separate marking established for the first time in an abstract framework, generalizes [C. Carstensen and E.-J. Park, SIAM J. Numer. Anal., 53 (2015), pp. 43-62] for least-squares schemes, and extends [C. Carstensen and H. Rabus, Math. Comp., 80 (2011), pp. 649-667] to the mixed finite element method with flux error control in H(div). The paper gives an outline of the mathematical analysis for optimal convergence rates but also serves as a reference so that future contributions merely verify a few axioms in a new application in order to ensure optimal mesh-refinement of the adaptive algorithm.

ANSYS Mechanical APDL for Finite Element Analysis provides a hands-on introduction to engineering analysis using one of the most powerful commercial general purposes finite element programs on the market.

In this paper, which is the second in a series of two, the preasymptotic error analysis of the continuous interior penalty finite element method (CIP-FEM) and the FEM for the Helmholtz equation in two and three dimensions is continued. While Part I contained results on the linear CIP-FEM and FEM, the present part deals with approximation spaces of order p ≥ 1. By using a modified duality argument, preasymptotic error estimates are derived for both methods under the condition of $\\frac{{kh}}{p} \\leqslant Co{(\\frac{p}{k})^{\\frac{1}{{p + 1}}}$ , where k is the wave number, h is the mesh size, and Co is a constant independent of k, h, p, and the penalty parameters. It is shown that the pollution errors of both methods in H¹-norm are O(k² p +¹h² p ) if p = O(1) and are O $(\\frac{k}{{{p^2}}}{(\\frac{{kh}}{{\\sigma p}})^{2p}})$ if the exact solution u ϵ H²(Ω) which coincide with existent dispersion analyses for the FEM on Cartesian grids. Here σ is a constant independent of k, h, p and the penalty parameters. Moreover, it is proved that the CIPFEM is stable for any k, h, p > 0 and penalty parameters with positive imaginary parts. Besides the advantage of the absolute stability of the CIP-FEM compared to the FEM, the penalty parameters may be tuned to reduce the pollution effects.