Gradient based enhanced finite element formulation for diffuse phase interfaces

Interfaces between adjacent phases, so-called domain walls, appear as non-linear gradients of order parameters in diffuse phase field models. Usually, the interface width is much smaller than the dimension of the simulated region. Since the position of domain walls is not known a priori the maximum size of finite elements needs to be adapted to the length scale of interfaces within the entire region. We suggested a selective finite element method to improve the numerical solution of diffuse phase field models [1, 2]. It enhances the finite element interpolation space using supplementary local degrees of freedom. However, corresponding additional nodes are strictly located in the interior of elements, thus, C⁰-continuity at element border is guaranteed. Since C⁰-continuity limits the performance of this method we propose in this paper a relaxation of C⁰-requirements perpendicular to the gradient of the order parameter. Therefore, the direction of interfaces is analyzed as additional information for further adaptive improvement of the interpolation space. A dual phase field model is used to validate the proposed method. The analytical solution of a stationary domain wall allows error analysis of regular and distorted finite element meshes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Collectionwise weak continuity duals

A function f: (X, τ) → (Y, σ) is weakly collectionwise continuous if for some C ? 2 ^X with τ ? C we have f ^?1(V) ∈ C for each V ∈ σ. In this case, f is said to be C-continuous. If also τ ? C* ? 2 ^X , C*-continuity is a dual to C-continuity if C?C* = τ and then the pair (C-continuity, C*-continuity) is a decomposition of continuity. In this paper, two natural topological methods are found for construction of a dual to any collectionwise weak continuity. Some known decompositions are improved.     

Error Estimates for Finite-Element Navier-Stokes Solvers without Standard Inf-Sup Conditions

The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C^1 elements for velocity and C^0 elements for pressure. A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag, Israeli, DeVille and Karniadakis.     

A compact C0 discontinuous Galerkin method for Kirchhoff plates

A compact C⁰ discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C⁰ finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C⁰ interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H¹‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015     

Iterative process for G-multi degree reduction of Bézier curves

In this paper, the issue of multi-degree reduction of Bézier curves with C¹ and G²-continuity at the end points of the curve is considered. An iterative method, which is the first of this type, is derived. It is shown that this algorithm converges and can be applied iteratively to get the required accuracy. Some examples and figures are given to demonstrate the efficiency of this method.     

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.     

广义强向量拟均衡问题组解的存在性

本文研究了一类集值广义强向量拟均衡问题组解的存在性问题.利用集值映射的自然拟C-凸性和集值映射的下(-C)-连续性的定义和Kakutani-Fan-Glicksberg不动点定理,在不要求锥C的对偶锥C~*具有弱*紧基的情况下,建立了该类集值广义强向量拟均衡问题组解的存在性定理.所得结果推广了该领域的相关结果.     

AC 0-collocation-like method for two-point boundary value problems

Summary A method of a collocation type based onC ⁰-piecewise polynomial spaces is presented for a two-point boundary value problem of the second order. The method has an optimal order of convergence under smoothness requirements on the exact solution which are weaker than forC ¹-collocation methods. If the differential operator is symmetric, a modification of this method leads to a symmetric system of linear equations. It is shown that if the collocation solution is a piecewise polynomial of degree not greater thanr, the method is stable and convergent with orderh ^r inH ¹-norm. A similar symmetric modification forC ⁰-colloction-finite element method [7] is also obtained. Superconvergence at the nodes is established.     

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.     

An adaptive <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>IPG method for the Helmholtz transmission eigenvalue problem

The interior penalty methods using C⁰ Lagrange elements (C⁰IPG) developed in the recent decade for the fourth order problems are an interesting topic in academia at present. In this paper, we discuss the adaptive fashion of C⁰IPG method for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal and dual eigenfunctions, and prove their reliability and efficiency. We also give the a posteriori error indicator for eigenvalues and design a C⁰IPG adaptive algorithm. Numerical experiments show that this algorithm is efficient and can get the optimal convergence rate.     

A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations

The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L^∞(L²) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L²(L²) norm for the first penalty method with a penalty parameter of order one (h^?1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001     

A stabilizer-free C0 weak Galerkin method for the biharmonic equations

In this article, we present and analyze a stabilizer-free C⁰ weak Galerkin(SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are C⁰ continuous piecewise polynomials of degree k + 2 with k≥0 and in terms of face unknowns which are discontinuous piecewise polynomials of degree k + 1. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the C¹ conformin...     

On an implicit particle method for simulation of forming processes

Task is the simulation of forming processes using particle methods. We implemented some mesh-free methods (the element free Galerkin method [1] and others) and the finite element method in one programme system which permits a direct comparison. For the mesh-free methods a moving least squares approximation is applied. The shape functions are not zero or one at the nodes, thus essential boundary conditions cannot be imposed directly [2]. We use a penalty method to enforce essential boundary conditions and contact conditions. The contact algorithm (normal contact of nodes to C¹-continuous surfaces) is checked by means of the element free Galerkin method and the FEM on the basis of numerical examples which deal with forming processes. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A Legendre spectral-element method for the one-dimensional fourth-order equations

A Legendre-Galerkin spectral-element method is proposed to solve the one-dimensional fourth-order equations. C¹-continuity between the elemental-faces is imposed by constructing appropriate basis functions. The method leads to linear systems with sparse matrices for the discrete variational formulations. Rigorous error analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results.     

Two-level additive Schwarz preconditioners for C 0 interior penalty methods

We study two-level additive Schwarz preconditioners that can be used in the iterative solution of the discrete problems resulting from C⁰ interior penalty methods for fourth order elliptic boundary value problems. We show that the condition number of the preconditioned system is bounded by C(1+(H³/δ³)), where H is the typical diameter of a subdomain, δ measures the overlap among the subdomains and the positive constant C is independent of the mesh sizes and the number of subdomains. This work was supported in part by the National Science Foundation under Grant No. DMS-03-11790.     

Penalty finite element approximations for the Stokes equations by L2 projection

This paper considers the penalty finite element method for the Stokes equations, based on some stable finite elements space pair (X_h, M_h) that do satisfy the discrete inf–sup condition. Theoretical results show that the penalty error converges as fast as one should expect from the order of the elements. Moreover, the penalty finite element method by L² projection can improve the penalty error estimates. Finally, we confirm these results by a series of numerical experiments. Copyright © 2008 John Wiley & Sons, Ltd.     

A Decomposition of a Weaker Form of Continuity

The aim of this paper is to give a decomposition of a weaker form of continuity, namely gα^**-continuity by providing the concepts of K-sets and K-continuity. This revised version was published online in June 2006 with corrections to the Cover Date.     

Smoothness of Stationary Subdivision on Irregular Meshes

We derive necessary and sufficient conditions for tangent plane and C ^k -continuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the universal surface . Any subdivision surface can be locally represented as a projection of the universal surface, which is uniquely defined by the subdivision scheme. This approach provides us with a more intuitive geometric understanding of subdivision near extraordinary vertices. February 16, 1998. Date revised: January 27, 1999. Date accepted: April 2, 1999.     

Efficient algorithms for solving tensor product finite element equations

Summary There are currently several highly efficient methods for solving linear systems associated with finite difference approximations of Poisson's equation in rectangular regions. These techniques are employed to develop both direct and iterative methods for solving the linear systems arising from the use ofC ⁰ quadratic orC ¹ cubic tensor product finite elements.