Elementary Finite Elements.- Nonconforming Finite Elements.- Mixed Finite Elements.- Discontinuous Finite Elements.- Characteristic Finite Elements.- Adaptive Finite Elements.- Solid Mechanics.- Fluid Mechanics.- Fluid Flow in Porous Media.- Semiconductor Modeling.

Finite element methods and their applications

Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

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Numerical methods for nonlocal and fractional models

In this paper, a new type of multi-level correction scheme is proposed forsolving eigenvalue problems by ﬁnite element method. With this new scheme,the accuracy of eigenpair approximations can be improved after each correc-tion step which only needs to solve a source problem on ﬁner ﬁnite elementspace and an eigenvalue problem on the coarsest ﬁnite element space. Thiscorrection scheme can improve the eﬃciency of solving eigenvalue problemsby ﬁnite element method.Keywords. Eigenvalue problem, multi-level correction scheme, ﬁnite ele-ment method, multi-space, multi-grid.AMS subject classiﬁcations. 65N30, 65B99, 65N25, 65L15 1 Introduction The purpose of this paper is to propose a new type of multi-level correction schemebased on ﬁnite element discretization to solve eigenvalue problems. The two-gridmethod for solving eigenvalue problems has been proposed and analyzed by Xu andZhou in [21]. The idea of the two-grid comes from [19, 20] for nonsymmetric orindeﬁnite problems and nonlinear elliptic equations. Since then, there have existedmany numerical methods for solving eigenvalue problems based on the idea of two-grid method ([1, 6, 17]).

/pdf/a-multi-level-correction-scheme-for-eigenvalue-problems-2b2h72be9e.pdf

A multi-level correction scheme for eigenvalue problems

The ubiquity of semilinear parabolic equations is clear from their numerous applications ranging from physics and biology to materials and social sciences. In this paper, we consider a practically ...

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Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes

A high-order and energy stable scheme is developed to simulate phase-field models by combining the semi-implicit spectral deferred correction (SDC) method and the energy stable convex splitting technique. The convex splitting scheme we use here is a linear unconditionally stable method but is only of first-order accuracy, so the SDC method can be used to iteratively improve the rate of convergence. However, it is found that the accuracy improvement may affect the overall energy stability which is intrinsic to the phase-field models. To compromise the accuracy and stability, a local $p$-adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level. The proposed strategy is found very useful for producing accurate numerical solutions at small time (dynamics) as well as long time (steady state) with reasonably large time stepsizes. Numerical experiments are carried out to demonstrate the high effectiveness of the proposed numerical strategy.

Long Time Numerical Simulations for Phase-Field Problems Using $p$-Adaptive Spectral Deferred Correction Methods

It has been demonstrated that spectral deferred correction (SDC) methods can achieve arbitrary high order accuracy and possess good stability properties. There have been some recent interests in using high-order Runge-Kutta methods in the prediction and correction steps in the SDC methods, and higher order rate of convergence is obtained provided that the quadrature nodes are uniform. The assumption of the use of uniform mesh has a serious practical drawback as the well-known Runge phenomenon may prevent the use of reasonably large number of quadrature nodes. In this work, we propose a modified SDC methods with high-order integrators which can yield higher convergence rates on both uniform and non-uniform quadrature nodes. The expected high-order of accuracy is theoretically verified and numerically demonstrated.

http://www.math.hkbu.edu.hk/~ttang/Papers/mar12-b.pdf

High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q
 1
 rot
 and EQ
 1
 rot
 . Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations.

/pdf/approximation-and-eigenvalue-extrapolation-of-stokes-3utjoot22w.pdf

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

The numerical solution of nonlocal constrained value problems with integrable kernels is considered. These nonlocal problems arise in nonlocal mechanics and nonlocal diffusion. The structure of the true solution to the problem is analyzed first. The analysis leads naturally to a new kind of discontinuous Galerkin method that can more efficiently solve the problem numerically. The new method is shown to be asymptotically compatible. Moreover, it has optimal convergence rate for any dimensional case under mild assumptions.

A Conforming DG Method for Linear Nonlocal Models with Integrable Kernels

Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods

In this paper, we analyze the biharmonic eigenvalue problem by two nonconforming finite elements, Q and E Q. We obtain full order convergence rate of the eigenvalue approximations for the biharmonic eigenvalue problem based on asymptotic error expansions for these two nonconforming finite elements. Using the technique of eigenvalue error expansion, the technique of integral identities, and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Xiaobo Yin

Papers

High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

A Conforming DG Method for Linear Nonlocal Models with Integrable Kernels

Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods

Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods