An optimal order multigrid method for biharmonic,C 1 finite element equations

Summary In this paper, we study a special multigrid method for solving large linear systems which arise from discretizing biharmonic problems by the Hsieh-Clough-Tocher,C ¹ macro finite elements or several otherC ¹ finite elements. Since the multipleC ¹ finite element spaces considered are not nested, the nodal interpolation operator is used to transfer functions between consecutive levels in the multigrid method. This method converges with the optimal computational order.     

A higher order finite element including transverse normal strain for linear elastic composite plates with general lamination configurations

This paper describes a higher-order global-local theory for thermal/mechanical response of moderately thick laminated composites with general lamination configurations. In-plane displacement fields are constructed by superimposing the third-order local displacement field to the global cubic displacement field. To eliminate layer-dependent variables, interlaminar shear stress compatibility conditions have been employed, so that the number of variables involved in the proposed model is independent of the number of layers of laminates. Imposing shear stress free condition at the top and the bottom surfaces, derivatives of transverse displacement are eliminated from the displacement field, so that C⁰ interpolation functions are only required for the finite element implementation. To assess the proposed model, the quadratic six-node C⁰ triangular element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate. Comparing to various existing laminated plate models, it is found that simple C⁰ finite elements with non-zero normal strain could produce more accurate displacement and stresses for thick multilayer composite plates subjected to thermal and mechanical loads. Finally, it is remarked that the proposed model is quite robust, such that the finite element results are not sensitive to the mesh configuration and can rapidly converge to 3-D elasticity solutions using regular or irregular meshes.     

Legendre-Laguerre coupled spectral element methods for second- and fourth-order equations on the half line

Some Legendre spectral element/Laguerre spectral coupled methods are proposed to numerically solve second- and fourth-order equations on the half line. The proposed methods are based on splitting the infinite domain into two parts, then using the Legendre spectral element method in the finite subdomain and Laguerre method in the infinite subdomain. C⁰ or C¹-continuity, according to the problem under consideration, is imposed to couple the two methods. Rigorous error analysis is carried out to establish the convergence of the method. More importantly, an efficient computational process is introduced to solve the discrete system. Several numerical examples are provided to confirm the theoretical results and the efficiency of the method.     

Semiregular Hermite Tetrahedral Finite Elements

Tetrahedral finite C ⁰-elements of the Hermite type satisfying the maximum angle condition are presented and the corresponding finite element interpolation theorems in the maximum norm are proved.     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.     

Stability of penalty finite-element methods for nonconforming problems

Penalty methods have been proposed as a viable method for enforcing interelement continuity constraints on nonconforming elements. Particularly for fourth-order problems in which C¹-continuity leads to elements of high degree or complex composite elements, the use of penalty methods to enforce the C¹-continuity constraint appears promising. In this study we demonstrate equivalence of the finite-element penalty method to a hybrid method and provide a stability analysis which implies that the penalty method is stable only if reduced integration of a certain order is used. Numerical experiments confirm that the penalty method fails if this condition is not met.     

What is the smallest possible constant in Céa’s lemma?

We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in ℝ^d with d ∈ {1, 2, 3, ...} The constant C ⩾ 1 appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and the Lagrange interpolant of the exact solution, we show that the ratio between discretization and interpolation errors is equal to as the discretization parameter h tends to zero. Numerical results in one and two-dimensional case illustrating this phenomenon are presented. This research was supported by Shandong Province Young Scientists Foundation of China 2005BS01008, Institutional Research Plan AV02 101 90503, and by Grant No A 1019201 of the Academy of Sciences of the Czech Republic.     

A note on the least squares solution of the groundwater flow equation

The least squares finite element method is a member of the weighted residuals class of numerical methods for solving partial differential equations. The least squares finite element method is applied to the groundwater flow equation. Space is discretized with a C¹ continuous trial function and parameters are approximated with a C⁰ bilinear basis. Solutions for problems containing parameters with large localized spatial gradients are characterized by errors that are propagated throughout the entire domain. Second-order spatial convergence is observed, and extreme mesh refinement is required to match Galerkin and mixed least squares finite element results. Temporal discretization should be kept separate from the least squares spatial discretization. © 1994 John Wiley & Sons, Inc.     

Triangular finite elements of HCT type and classC ρ

Let τ be some triangulation of a planar polygonal domain Ω. Given a smooth functionu, we construct piecewise polynomial functionsv∈C ^ρ(Ω) of degreen=3 ρ for ρ odd, andn=3ρ+1 for ρ even on a subtriangulation τ₃ of τ. The latter is obtained by subdividing eachT∈ρ into three triangles, andv/T is a composite triangular finite element, generalizing the classicalC ¹ cubic Hsieh-Clough-Tocher (HCT) triangular scheme. The functionv interpolates the derivatives ofu up to order ρ at the vertices of τ. Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements of this type.     

C 1 Quadratic and C 2 Quartic Macro-Elements on a Modified Morgan-Scott Triangulation

In this paper, we construct a quadratic composite finite element of class C ¹ and quartic composite finite element of class C ² on a new triangulation τ ₁₀ which is obtained by splitting each triangle of a given triangulation τ into ten smaller subtriangles. These new elements can be used for constructing spline spaces with local basis that can be applied for solving some Hermite interpolation problems with optimal approximation order.     

Sharp estimates for finite element approximations to parabolic problems with Neumann boundary data of low regularity

Consider a homogeneous parabolic problem on a smooth bounded domain in ℝ ^N but with initial data and Neumann boundary data of low regularity. Sharp interior maximum norm error estimates are given for a semidiscrete C ⁰ finite element approximation to this problem. These estimates are obtained by first establishing a new localized L ^∞ estimate for semidiscrete finite element approximations on interior subdomains. Numerical examples illustrate the findings. AMS subject classification (2000) 65N30     

Morley type non‐C0 nonconforming rectangular plate finite elements on anisotropic meshes

In this article, two Morley type non‐C⁰ nonconforming rectangular finite elements are discussed to numerically solve the fourth order plate bending problem under anisotropic meshes. The optimal anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches. Some numerical tests are given to confirm the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Structural contact problems for large deformations - Comparison of steady and non steady normal field

We will present a comparison between two formulations of the normal vector field for contact algorithms based on the mortar method. First the non steady normal field is discussed. The non steadiness is a result of the C⁰ continuity of the boundary discretization. This is the common result if one discretize the domain with classical finite element methods. Second we will present results for a special normal field distribution. We average the nodal normal vector of two ascending edges and interpolate this nodal normal throughout the edges. We have implemented both methods and present comparisons based on numerical experiments. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Meshes for Finite Elements of Arbitrary Order

Given a function f defined on a bounded domain Ω⊂ℝ² and a number N>0, we study the properties of the triangulation T_N\mathcal{T}_{N} that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most N elements. The error is studied in the norm X=L ^p for 1≤p≤∞, and we consider Lagrange finite elements of arbitrary polynomial degree m−1. We establish sharp asymptotic error estimates as N→+∞ when the optimal anisotropic triangulation is used, recovering the results on piecewise linear interpolation (Babenko et al. in East J. Approx. 12(1), 71–101, 2006; Babenko, submitted; Chen et al. in Math. Comput. 76, 179–204, 2007) and improving the results on higher degree interpolation (Cao in SIAM J. Numer. Anal. 45(6), 2368–2391, 2007, SIAM J. Sci. Comput. 29, 756–781, 2007, Math. Comput. 77, 265–286, 2008). These estimates involve invariant polynomials applied to the m-th order derivatives of f. In addition, our analysis also provides practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant. We partially extend our results to higher dimensions for finite elements on simplicial partitions of a domain Ω⊂ℝ ^d .     

Variational Problems in Domains with Cusp-Points and the Finite Element Method

A variational problem in a two-dimensional domain with cusp-points corresponding to a linear elliptic boundary value problem is formulated and the unique existence of its solution is proved. The corresponding finite element method using triangular finite C ⁰-elements with polynomials of the first degree is analyzed and both the convergence (under the assumptions sufficient for the existence of the exact solution) and the maximal rate of convergence 𝒪(h) are proved.     

Equivariant Surgery Theory: Deleting-Inserting Theorems of Fixed Point Manifolds on Spheres and Disks

The paper gives a tool to delete and insert fixed point manifolds for smooth actions of finite Oliver groups on spheres and disks. A similar result was already given in a joint article with E. Laitinen and K. Pawaowski for those of finite nonsolvable groups on spheres. It is useful in classifying smooth actions on spheres from the view point of fixed point data. The methods employed in the present paper are equivariant surgery and equivariant connected sum associated with elements in the Burnside ring. The idea of killing surgery obstructions is as follows: Let G be a finite group not of prime power order, C a contractible, finiteG CW complex, and an element in a K theoretic group arising as an obstruction class of geometric object f. It often holds that (1-[C])^m becomes trivial for large integers m where [C] is the element represented byC in the Burnside ring (G). One expects that the algebraic object (1 - [C])^m is realizable as the obstruction class of G connected sum of f's related to (1 - [C])^m. Since it is true for the case here, we can kill the obstruction by taking G connected sum off's     

Interpolating minimal energy C1‐Surfaces on Powell–Sabin Triangulations: Application to the resolution of elliptic problems

In this article, we present a method to obtain a C¹‐surface, defined on a bounded polygonal domain Ω, which interpolates a specific dataset and minimizes a certain “energy functional.” The minimization space chosen is the one associated to the Powell–Sabin finite element, whose elements are C¹‐quadratic splines. We develop a general theoretical framework for that, and we consider two main applications of the theory. For both of them, we give convergence results, and we present some numerical and graphical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 798–821, 2015     

The interpolation problem for ridge functions

It is shown that the interpolation problem for ridge functions can be solved if and only if the rank of a certain matrix A equals the number of interpolation points. The elements of the matrix A are either 0 or 1 and can be easilyfound from the arguments of the unknown functions. It is shown that Sun's Characteristic, or incidence matrix C is given by C = AA ^T . From this it follows that the rank condition is equivalent to Sun's positive definite C condition.     

The least squares spectral element method for the Cahn–Hilliard equation

The problem of numerically resolving an interface separating two different components is a common problem in several scientific and engineering applications. One alternative is to use phase field or diffuse interface methods such as the Cahn–Hilliard (C–H) equation, which introduce a continuous transition region between the two bulk phases. Different numerical schemes to solve the C–H equation have been suggested in the literature. In this work, the least squares spectral element method (LS-SEM) is used to solve the Cahn–Hilliard equation. The LS-SEM is combined with a time–space coupled formulation and a high order continuity approximation by employing C¹¹p-version hierarchical interpolation functions both in space and time. A one-dimensional case of the Cahn–Hilliard equation is solved and the convergence properties of the presented method analyzed. The obtained solution is in accordance with previous results from the literature and the basic properties of the C–H equation (i.e. mass conservation and energy dissipation) are maintained. By using the LS-SEM, a symmetric positive definite problem is always obtained, making it possible to use highly efficient solvers for this kind of problems. The use of dynamic adjustment of number of elements and order of approximation gives the possibility of a dynamic meshing procedure for a better resolution in the areas close to interfaces.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes

An L ²-estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.