Energy Error Estimates Research Articles

A Nonconforming Finite Element Approximation for the von Karman equations

In this paper, a nonconforming finite element method has been proposed and analyzed for the von Karman equations that describe bending of thin elastic plates. Optimal order error estimates in broken energy and H 1 norms are derived under minimal regularity assumptions. Numerical results that justify the theoretical results are presented.

Conforming finite element methods for the von Kármán equations

In this paper, we consider canonical von Karman equations that describe the bending of thin elastic plates defined on polygonal domains. A conforming finite element method is employed to approximate the displacement and Airy stress functions. Optimal order error estimates in energy, H1 and L2 norms are deduced. The results of numerical experiments confirm the theoretical results obtained.

Guaranteed Energy Error Estimators for a Modified Robust Crouzeix–Raviart Stokes Element

This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix---Raviart finite element method for the Stokes equations. The modification from Linke (Comput Methods Appl Mech Eng 268:782---800, 2014) is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method.

Finite-element harmonic experiments to model fractured induced anisotropy in poroelastic media

Fractures in a fluid-saturated poroelastic–Biot–medium can be modeled as very thin highly permeable and compliant layers within a porous background. A Biot medium containing a dense set of aligned fractures behaves as an effective transversely isotropic and viscoelastic (TIV) medium at the macroscale when the predominant wavelength is much larger than the average distance between fractures. One important mechanism in Biot media at seismic frequencies is wave-induced fluid flow generated by fast compressional waves at mesoscopic-scale heterogeneities, generating slow diffusion-type Biot waves. In this work, we present and analyze a collection of time-harmonic finite element experiments that take into account the effects of the presence of aligned fractures and interlayer fluid flow occurring at the mesoscale, allowing us to determine the complex and frequency dependent stiffnesses of the effective TIV medium at the macroscale.These numerical upscaling experiments are defined as boundary value problems on representative samples of the fractured material, with boundary conditions associated with compressibility and shear tests, which are solved using the finite element (FE) method. The FE space chosen to discretize each component of the solid displacement vector is that of globally continuous piecewise bilinear functions, while for the fluid phase the vector part of the Raviart–Thomas–Nedelec space of zero order is employed. We present results on the uniqueness of the solution of the continuous and discrete problems, and derive optimal a priori energy error estimates. First, the numerical results are validated with those of a theory valid for fluid flow perpendicular to the fracture layering and independent of the loading direction, so that the attenuation mechanism can be represented by a single relaxation function. Then, the methodology is applied to cases for which no analytical solutions are available, such as a fractured Biot medium saturated with brine and patches of CO2 and a brine saturated sample of uniform background and fractures with fractal variations in their petrophysical properties.

A new cubic nonconforming finite element on rectangles

A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the 12 values at the three Gauss points on each of the four edges. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are 11. The nonconforming element consists of . We count the corresponding dimension for Dirichlet and Neumann boundary value problems of second‐order elliptic problems. We also present the optimal error estimates in both broken energy and norms. Finally, numerical examples match our theoretical results very well. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 691–705, 2015

Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal $L^2$-error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes.

New error estimates of nonconforming mixed finite element methods for the Stokes problem

In this paper, we revisit the classical error estimates of nonconforming Crouzeix–Raviart type finite elements for the Stokes equations. By introducing some quasi-interpolation operators and using the special properties of these nonconforming elements, it is proved that their consistency errors can be bounded by their approximation errors together with a high-order term, especially which can be of arbitrary order provided that f in the right-hand side is piecewise smooth enough. Furthermore, we show an interesting result that both in the energy norm and L2 norm the consistency errors are dominated by the approximation errors of their finite element spaces. As byproducts, we derive the error estimates in both energy and L2 norms under the regularity assumption (u,p) ∈ H1 + s(Ω) × Hs(Ω) with any s ∈ (0,1], which fills the gap in the a priori error estimate of these nonconforming elements with low regularity . Furthermore, a robust convergence is proved with minimal regularity assumption s = 0. These results seem to be missing in the literature. Numerical tests are provided, confirming the analysis, especially the new results on the L2 convergence. Copyright © 2013 John Wiley & Sons, Ltd.

Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra

The aim of the paper is to show the stability of the finite element solution for the Stokes system in W ∞ 1 W^1_\infty norm on general convex polyhedral domain. In contrast to previously known results, W r 2 W^2_r regularity for r > 3 r>3 , which does not hold for general convex polyhedral domains, is not required. The argument uses recently available sharp Hölder pointwise estimates of the corresponding Green’s matrix together with novel local energy error estimates, which do not involve an error of the pressure in a weaker norm.

Error estimates in the projection-difference method for a hyperbolic-parabolic system of abstract differential equations

We consider a Cauchy problem for a hyperbolic-parabolic system of abstract differential equations in a Hilbert space, which generalizes a number of linear coupled thermoelasticity problems. Energy error estimates for a projection-difference method as applied to the Cauchy problem are established without imposing any special conditions on projection subspaces.

Local energy estimates for the finite element method on sharply varying grids

Local energy error estimates for the nite element method for el- liptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local ap- proximation term, plus a global \pollution term that measures the inuence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which much more closely matches the typical practical situation. Our chief technical innovation is an improved superapproximation result.

On the error bounds of nonconforming finite elements

We prove that the error estimates of a large class of nonconforming finite elements are dominated by their approximation errors, which means that the well-known Cea’s lemma is still valid for these nonconforming finite element methods. Furthermore, we derive the error estimates in both energy and L 2 norms under the regularity assumption u ∈ H 1+s (Ω) with any s > 0. The extensions to other related problems are possible.

Asynchronous multi-domain variational integrators for nonlinear hyperelastic solids

We present the asynchronous multi-domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary-value problem in hyperelasticity. Variational time integration schemes, based on the principle of minimal action within the Lagrangian framework, are constructed for the equation of motion and implemented into a variational finite element framework, which is systematically derived from the three-field de Veubeke-Hu-Washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly-incompressible materials. For efficient parallel computing, we use the dual domain decomposition method with local Lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains. The α-method for time discretization and the multi-domain spatial decomposition enable us to use different types of integrators (explicit vs. implicit) and different time steps on different parts of a computational domain, and thus efficiently capture the underlying physics with less computational effort. The energy conservation of our nonlinear, midpoint, asynchronous integration scheme is investigated using the Energy method, and both local and the global energy error estimates are derived. We illustrate the performance of proposed variational multi-domain time integrators by means of three examples. First, the method of manufactured solutions is used to examine the consistency of the formulation. In the second example, we investigate energy conservation and stability. Finally, we apply the method to the motion of a heterogeneous plane domain, where different integrators and time discretization steps are used accordingly with disparate material data of individual parts.

Sparse Tensor Product Wavelet Approximation of Singular Functions

On product domains, sparse-grid approximation yields optimal, dimension-independent convergence rates when the function that is approximated has $L^2$-bounded mixed derivatives of a sufficiently high order. We show that the solution of Poisson's equation on the n-dimensional hypercube with Dirichlet boundary conditions and smooth right-hand side generally does not satisfy this condition. As suggested by P.-A. Nitsche in [Constr. Approx., 21 (2005), pp. 63–81], the regularity conditions can be relaxed to corresponding ones in weighted $L^2$ spaces when the sparse-grid approach is combined with local refinement of the set of one-dimensional wavelet indices towards the end points. In this paper, we prove that for general smooth right-hand sides, the solution of Poisson's problem satisfies these relaxed regularity conditions in any space dimension. Furthermore, since we remove log-factors from the energy-error estimates from Nitsche's work, we show that in any space dimension, locally refined sparse-grid approximation yields the optimal, dimension-independent convergence rate.

Analysis for quadrilateral MITC elements for the Reissner-Mindlin plate problem

The present paper is made up of two parts. In the first part, we study the mathematical stability and convergence of the quadrilateral MITC elements for the Reissner-Mindlin plate problem in an abstract setting. We generalize the Brezzi-Bathe-Fortin conditions to the quadrilateral MITC elements by weakening the second and fourth conditions. Under these conditions, we show the well-posedness of the discrete problem and establish an abstract error estimate in the energy norm. The conclusion of this part is sparsity in the mathematical research of the quadrilateral MITC elements in the sense that one only needs to check these five conditions. In the second part, we extend four families of rectangular MITC elements of Stenberg and Suri to the quadrilateral meshes. We prove that these quadrilateral elements satisfy the generalized Brezzi-Bathe-Fortin conditions from the first part. We develop the h-p error estimates in both energy and L 2 norm for these quadrilateral elements. For the first three families of quadrilateral elements, the error estimates indicate that their convergent rates in both energy and L 2 norm depend on the mesh distortion parameter a. We can get optimal error estimates for them provided that a = 1. In addition, we show the optimal convergence rates in energy norm uniformly in a for the fourth family of quadrilateral elements. Like their rectangular counterparts, these quadrilateral elements are locking-free.

Error estimates for schemes of the projection-difference method for abstract quasilinear hyperbolic equations

We study the convergence of the three-layer scheme of the projection-difference method for abstract quasilinear hyperbolic equations in Hilbert space. We establish asymptotic energy error estimates for an arbitrary choice of finite-dimensional subspaces in which the approximation problems are solved.

Analysis of a Nonsymmetric Discontinuous Galerkin Method for Elliptic Problems: Stability and Energy Error Estimates

In this paper we analyze a nonsymmetric discontinuous Galerkin method for elliptic problems proposed by Oden, Babuska, and Baumann. Our main results are a complete inf-sup stability analysis and, as a consequence, error estimates in a mesh dependent energy norm allowing variable meshsize and order of polynomials. The analysis is carried out in two spatial dimensions on an unstructured triangulation.

On the Finite Volume Element Method for General Self-Adjoint Elliptic Problems

The finite volume element method (FVE) is a discretization technique for partial differential equations. This paper develops discretization energy error estimates for general self-adjoint elliptic boundary value problems with FVE based on triangulations, on which there exist linear finite element spaces, and a very general type of control volumes (covolumes). The energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in [R. E. Bank and D. J. Rose, SIAM J. Numer. Anal., 24 (1987), pp. 777--787], [Z. Q. Cai, Numer. Math., 58 (1991), pp. 713--735] are removed. The authors finally provide a counterexample to show that an expected L2-error estimate does not exist in the usual sense. It is conjectured that the optimal order of $\|u-u_h\|_{0,\Omega}$ should be O(h) for the general case.

On energy error estimates of the domain decomposition method with penalty in solving parabolic problems

On energy error estimates of the domain decomposition method with penalty in solving parabolic problems

Finite Element Error Estimates for a Blast Furnace Gas Flow Problem

In the mathematical modeling of gas flow in an iron-making blast furnace, a nonlinear second-order boundary value problem is obtained. The operator associated with this problem is monotone and continuous. This paper shows how the algebraic properties of the principal nonlinear coefficient may be exploited to derive optimal order energy error estimates for the finite element approximations of this problem. The situation where the coefficient is implicitly defined is also considered.

A stable extrapolation method for multidimensional degenerate parabolic problems

Degenerate parabolic problems in several space variables are approximated by combining a preliminary regularization procedure with a finite element extrapolation method. The proposed extrapolation acts on the so-called phase variable and leads to a linear problem which is shown to be stable. The ensuing linear algebraic system involves the same matrix for all time steps. Energy error estimates are also derived for the physical unknowns. An O ( h 1 / 2 ) O({h^{1/2}}) rate of convergence is proved, provided the approximation parameters are suitably related. In case the linear systems are solved by an iterative algorithm, such as the conjugate gradient method, an O ( h 3 / 2 ) O({h^{3/2}}) tolerance for the error reduction is shown to preserve the overall accuracy; the required computational effort is thus nearly optimal.