Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.     

High-order finite element methods for time-fractional partial differential equations

The aim of this paper is to develop high-order methods for solving time-fractional partial differential equations. The proposed high-order method is based on high-order finite element method for space and finite difference method for time. Optimal convergence rate O((Δt)^2−α+N^−r) is proved for the (r−1)th-order finite element method (r≥2).     

New adaptive mixed finite element method (AMFEM)

The need to develop reliable and efficient adaptive algorithms using mixed finite element methods arises from various applications in fluid dynamics and computational continuum mechanics. In order to save degrees of freedom, not all but just some selected set of finite element domains are refined and hence the fundamental question of convergence requires a new mathematical argument as well as the question of optimality. We will present a new adaptive algorithm for mixed finite element methods to solve the model Poisson problem, for which optimal convergence can be proved. The a posteriori error control of mixed finite element methods dates back to Alonso (1996) Error estimators for a mixed method. and Carstensen (1997) A posteriori error estimate for the mixed finite element method. The error reduction and convergence for adaptive mixed finite element methods has already been proven by Carstensen and Hoppe (2006) Error Reduction and Convergence for an Adaptive Mixed Finite Element Method, Convergence analysis of an adaptive nonconforming finite element methods. Recently, Chen, Holst and Xu (2008) Convergence and Optimality of Adaptive Mixed Finite Element Methods. presented convergence and optimality for adaptive mixed finite element methods following arguments of Rob Stevenson for the conforming finite element method. Their algorithm reduces oscillations, before applying and a standard adaptive algorithm based on usual error estimation. The proposed algorithm does this in a natural way, by switching between the reduction of either the estimated error or oscillations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B‐spline collocation method

In the present study, the operator splitting techniques based on the quintic B‐spline collocation finite element method are presented for calculating the numerical solutions of the Rosenau–KdV–RLW equation. Two test problems having exact solutions have been considered. To demonstrate the efficiency and accuracy of the present methods, the error norms L₂ and L_∞ with the discrete mass Q and energy E conservative properties have been calculated. The results obtained by the method have been compared with the exact solution of each problem and other numerical results in the literature, and also found to be in good agreement with each other. A Fourier stability analysis of each presented method is also investigated.     

The Finite Cell Method for large deformation analysis

The Finite Cell Method (FCM) combines the fictitious domain approach with high-order finite elements and adaptive integration. For linear elastic problems with smooth solution, FCM has been shown to achieve exponential rates of convergence in energy norm, while its structured cell grid guarantees simple mesh generation irrespective of the geometric complexity involved. In this contribution, the FCM idea is combined with standard finite element technology for the solution of geometrically nonlinear problems. In particular, a modified FCM formulation is introduced, which resets the deformed configuration of the fictitious domain to the deformation-free reference configuration after each Newton iteration. Numerical experiments show that this intervention allows for stable nonlinear FCM analysis with very small values of the penalty parameter, while the accuracy of the geometrically nonlinear solution within the physical domain remains unaffected. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the convergence and superconvergence for a class of two-dimensional time fractional reaction-subdiffusion equations

This paper analyzes a class of two-dimensional (2-D) time fractional reaction-subdiffusion equations with variable coefficients. The high-order L2-1_σ time-stepping scheme on graded meshes is presented to deal with the weak singularity at the initial time t = 0, and the bilinear finite element method (FEM) on anisotropic meshes is used for spatial discretization. Using the modified discrete fractional Grönwall inequality, and combining the interpolation operator and the projection operator, the L²-norm error estimation and H¹-norm superclose results are rigorously proved. The superconvergence result in the H¹-norm is derived by applying the interpolation postprocessing technique. Finally, numerical examples are presented to verify the validation of our theoretical analysis.     

A characteristics-mixed finite element method for Burgers’ equation

In this paper, we propose a new mixed finite element method, called the characteristics-mixed method, for approximating the solution to Burgers’ equation. This method is based upon a space-time variational form of Burgers’ equation. The hyperbolic part of the equation is approximated along the characteristics in time and the diffusion part is approximated by a mixed finite element method of lowest order. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Our analysis show the new method approximate the scalar unknown and the vector flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the new scheme is easily implemented, shocks and boundary layers are handled with almost no oscillations. One of the contributions of the paper is to show how the optimal error estimates inL ²(Ω) are obtained which are much more difficult than in the standard finite element methods. These results seem to be new in the literature of finite element methods.     

Numerical analysis of a modified finite element nonlinear Galerkin method

Summary. A fully discrete modified finite element nonlinear Galerkin method is presented for the two-dimensional equation of Navier-Stokes type. The spatial discretization is based on two finite element spaces X_H and X_h defined on a coarse grid with grid size H and a fine grid with grid size h << H, respectively; the time discretization is based on the Euler explicit scheme with respect to the nonlinear term. We analyze the stability and convergence rate of the method. Comparing with the standard finite element Galerkin method and the nonlinear Galerkin method, this method can admit a larger time step under the same convergence rate of same order. Hence this method can save a large amount of computational time. Finally, we provide some numerical tests on this method, the standard finite element Galerkin method, and the nonlinear Galerkin method, which are in a good agreement with the theoretical analysis.Mathematics Subject Classification (2000): 35Q30, 65M60, 65N30, 76D05     

Probabilistic nonlinear analysis of CFR dams by MCS using Response Surface Method

This paper presents the probabilistic analysis of concrete-faced rockfill (CFR) dams according to the Monte Carlo Simulation (MCS) results which are obtained through the Response Surface Method (RSM). ANSYS finite element program is used to get displacement and principal stress components. First of all, some parametric studies are performed according to the simple and representative finite element model of dam body to obtain the optimum approximate model. Secondly, a sensitivity analysis is performed to get the most effective parameters on dam response. Then, RSM is used to obtain the approximate function through the selected parameters. After the performed analyses, star experimental design with quadratic function without mixed terms according to the k = 1 is determined as the most appropriate model. Finally, dam-foundation-reservoir interaction finite element model is constituted and probabilistic analyses are performed with MCS using the selected parameters, sampling method, function and arbitrary factor under gravity load for empty and full reservoir conditions. Geometrically and materially nonlinearity are considered in the analysis of dam-foundation-reservoir interaction system. Reservoir water is modeled by fluid finite elements based on the Lagrangian approach. Structural connections are modeled as welded contact and friction contact based on Coulomb’s friction law. Probabilistic displacements and stresses are presented and compared with deterministic results.     

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

In this work, an effective and fast finite element numerical method with high-order accuracy is discussed for solving a nonlinear time fractional diffusion equation. A two-level linearized finite element scheme is constructed and a temporal–spatial error splitting argument is established to split the error into two parts, that is, the temporal error and the spatial error. Based on the regularity of the time discrete system, the temporal error estimate is derived. Using the property of the Ritz projection operator, the spatial error is deduced. Unconditional superclose result in H¹-norm is obtained, with no additional regularity assumption about the exact solution of the problem considered. Then the global superconvergence error estimate is obtained through the interpolated postprocessing technique. In order to reduce storage and computation time, a fast finite element method evaluation scheme for solving the nonlinear time fractional diffusion equation is developed. To confirm the theoretical error analysis, some numerical results are provided.     

Improved L2 and H1 error estimates for the Hessian discretization method

The Hessian discretization method (HDM) for fourth-order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit-conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods (ncFEMs), finite volume methods (FVMs) and methods based on gradient recovery operators. A generic error estimate has been established in L², H¹, and H²-like norms in literature. In this paper, we establish improved L² and H¹ error estimates in the framework of HDM and illustrate it on various schemes. Since an improved L² estimate is not expected in general for FVM, a modified FVM is designed by changing the quadrature of the source term and a superconvergence result is proved for this modified FVM. In addition to the Adini ncFEM, in this paper, we show that the Morley ncFEM is an example of HDM. Numerical results that justify the theoretical results are also presented.     

A posteriori error analysis for eigenvalue problems

A new a posteriori error analysis for symmetric eigenvalue problems of the adaptive finite element method (AFEM) is presented. Based on an H¹ stable L² projection, we prove reliability of the edge contribution for P₁ finite element methods. Hence the AFEM proposed can ignore the volume contribution as well as oscillations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.     

Superconvergence of the - version of the finite element method in one dimension

In this paper, we investigate the superconvergence properties of the h-p version of the finite element method (FEM) for two-point boundary value problems. A postprocessing technique for the h-p finite element approximation is analyzed. The analysis shows that the postprocess improves the order of convergence. Furthermore, we obtain asymptotically exact a posteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis.     

A space–time discontinuous Galerkin method for linear convection-dominated Sobolev equations

This article presents a space–time discontinuous Galerkin (DG) finite element method for linear convection-dominated Sobolev equations. The finite element method has basis functions that are continuous in space and discontinuous in time, and variable spatial meshes and time steps are allowed. In the discrete intervals of time, using properties of the Radau quadrature rule, eliminates the restriction to space–time meshes of convectional space–time Galerkin methods. The existence and uniqueness of the approximate solution are proved. An optimal priori error estimate in L^∞(H¹) is derived. Numerical experiments are presented to confirm theoretical results.     

Structural Optimization with FEM-based Shakedown Analyses

In this paper, a mathematical programming formulation is presented for the structural optimization with respect to the shakedown analysis of 3-D perfectly plastic structures on basis of a finite element discretization. A new direct algorithm using plastic sensitivities is employed in solving this optimization formulation. The numerical procedure has been applied to carry out the shakedown analysis of pipe junctions under multi-loading systems. The new approach is compared to so-called derivative-free direct search methods. The computational effort of the proposed method is much lower compared to this methods.     

A space‐time mixed‐hybrid finite element method for the damped wave equation

A space‐time finite element method is introduced to solve the linear damped wave equation. The scheme is constructed in the framework of the mixed‐hybrid finite element methods, and where an original conforming approximation of H(div;Ω) is used, the latter permits us to obtain an upwind scheme in time. We establish the link between the nonstandard finite difference scheme recently introduced by Mickens and Jordan and the scheme proposed. In this regard, two approaches are considered and in particular we employ a formulation allowing the solution to be marched in time, i.e., one only needs to consider one time increment at a time. Numerical results are presented and compared with the analytical solution illustrating good performance of the present method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Penalty-combined approaches to the Ritz-Galerkin and finite element methods for singularity problems of elliptic equations

Penlty coupling techniques on an interface boundary, artificial or material, are first presented for combining the Ritz–Galerkin and finite element methods. An optimal convergence rate first is proved in the Sobolev norms. Moreover, a significant coupling strategy, L + 1 = O(|ln h|), between these two methods are derived for the Laplace equation with singularities, where L + 1 is the total number of particular solutions used in the Ritz–Galerkin method, and h is the maximal boundary length of quasiuniform elements used in the linear finite element method. Numreical experiments have been carried out for solving the benchmark model: Motz's problem. Both theoretical analysis and numreical experiments clearly display the importance of penalty-combined methods is solving elliptic equations with singularities.     

Applications of Lobatto polynomials to an adaptive finite element method: A posteriori error estimates for HP-Adaptivity and Grid-to-Grid Interpolation

Summary.  Hp-adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions between grids. Lobatto polynomial-based techniques are presented for both. An interpolation error-based error estimation strategy for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution. The estimates involve high-order derivatives of the solution that must be approximated from the computed solution. Differentiating a ``Taylor-like' series for error in the Lobatto interpolant and using the weak form of the equations yields the correct derivative approximations. This leads to a more robust order selection strategy. Interpolation between grids is done over each element using the Lobatto interpolating polynomial. Explicit formulas for the inverse of the resulting Lobatto interpolation matrices are given. Computational results illustrate the theory. Received June 25, 2001 / Revised version received February 12, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65M15,65M20,65M60 This research was partially supported by NSF Grant #DMS-0196108.     

On some versions of the element agglomeration AMGe method

The present paper deals with element‐based algebraic multigrid (AMGe) methods that target linear systems of equations coming from finite element discretizations of elliptic partial differential equations. The individual element information (element matrices and element topology) is the main input to construct the AMG hierarchy. We study a number of variants of the spectral agglomerate AMGe method. The core of the algorithms relies on element agglomeration utilizing the element topology (built recursively from fine to coarse levels). The actual selection of the coarse degrees of freedom is based on solving a large number of local eigenvalue problems. Additionally, we investigate strategies for adaptive AMG as well as multigrid cycles that are more expensive than the V‐cycle utilizing simple interpolation matrices and nested conjugate gradient (CG)‐based recursive calls between the levels. The presented algorithms are illustrated with an extensive set of experiments based on a matlab implementation of the methods. Copyright © 2008 John Wiley & Sons, Ltd.