Numerical solution of the Fokker Planck equation using moving finite elements

Numerical solution of the Fokker Planck equation for the probability density function of a stochastic process by traditional finite difference or finite element methods produces erroneous oscillations and negative values whenever the drift is large compared to the diffusion. Upwinding schemes to eliminate the oscillations introduce false numerical diffusion because it is impossible to make the one step drift large enough to match the original equation without making the one step diffusion too large. A variation of the moving finite element method is presented that overcomes these difficulties by using basis functions that satisfy the drift part of the equation by moving along the trajectories of the deterministic dynamical system associated with the stochastic process. A Galerkin type method can then be used to find the coefficients in the remaining pure diffusion equation. Solutions of two test equations are presented to illustrate the effectiveness of the method.     

Operator splitting for numerical solution of the modified Burgers' equation using finite element method

The aim of this study is to obtain numerical behavior of a one‐dimensional modified Burgers' equation using cubic B‐spline collocation finite element method after splitting the equation with Strang splitting technique. Moreover, the Ext4 and Ext6 methods based on Strang splitting and derived from extrapolation have also been applied to the equation. To observe how good and effective this technique is, we have used the well‐known the error norms L₂ and L_∞ in the literature and compared them with previous studies. In addition, the von Neumann (Fourier series) method has been applied after the nonlinear term has been linearized to investigate the stability of the method.     

Weak Galerkin finite element method for a class of time fractional generalized Burgers' equation

In this article, we use the weak Galerkin (WG) finite element method to study a class of time fractional generalized Burgers' equation. The existence of numerical solutions and the stability of fully discrete scheme are proved. Meanwhile, by applying the energy method, an optimal order error estimate in discrete L² norm is established. Numerical experiments are presented to validate the theoretical analysis.     

The finite element method with nodes moving along the characteristics for convection-diffusion equations

Summary We examine the convergence properties of the finite element method with nodes moving along the characteristics for one-dimensional convection-diffusion equations. For linear elements, we demonstrate optimal rates of convergence in theL ²,H ¹ andL norms. Both linear and nonlinear problems are considered.This work forms part of the research programme of the Oxford/Reading Institute for Computational Dynamics.     

On the least-squares conjugate-gradient solution of the finite element approximation of Burgers' equation

A finite element approximation of the two-dimensional steady Burgers' equation is presented and a conjugate gradient approach is taken to solve the resulting finite element equations. The scheme is computationally efficient and is relatively easy to implement. An optimal error bound is established and a set of test problems with known analytic solutions is given to demonstrate the efficiency of the method.     

Finite-parameter feedback stabilization of original Burgers' equations and Burgers' equation with nonlocal nonlinearities

We study the problem of global exponential stabilization of original Burgers' equations and the Burgers' equation with nonlocal nonlinearities by controllers depending on finitely many parameters. We investigate both equations by employing controllers based on finitely many Fourier modes and the latter equation by employing finitely many volume elements. To ensure global exponential stabilization, we have provided sufficient conditions on the control parameters for each problem. We also show that solutions of the controlled equations are steering a concrete solution of the non-controlled system as t → ∞ with an exponential decay rate.     

Higher-order finite elements with mass-lumping for the 1D wave equation

This paper is devoted to the construction and analysis of a method, higher order in space and time, for solving the one-dimensional wave equation. This method is based on P₃ Lagrange finite elements with mass-lumping which avoids the inversion of a mass matrix at each time-step. The mass-lumping implies to make the abscissae of the interior points coincide with these of the Gauss-Lobatto quadrature rule. A Fourier analysis of the method for a regular mesh points out a superconvergence result. The gain of accuracy is illustrated by numerical experiments.     

Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number

A fourth-order compact finite difference scheme is employed with the multigrid algorithm to obtain highly accurate numerical solution of the convection-diffusion equation with very high Reynolds number and variable coefficients. The multigrid solution process is accelerated by a minimal residual smoothing (MRS) technique. Numerical experiments are employed to show that the proposed multigrid solver is stable and yields accurate solution for high Reynolds number problems. We also show that the MRS acceleration procedure is efficient and the acceleration cost is negligible. © 1997 John Wiley & Sons, Inc.     

Numerical solution of RLW equation using linear finite elements within Galerkin's method

The regularised long wave equation is solved by Galerkin's method using linear space finite elements. In the simulations of the migration of a single solitary wave, this algorithm is shown to have good accuracy for small amplitude waves. Moreover, for very small amplitude waves (⩽0.09) it has higher accuracy than an approach using quadratic B-spline finite elements within Galerkin's method. The interaction of two solitary waves is modelled for small amplitude waves.     

A formula for the error of finite sinc interpolation with an even number of nodes

Sinc interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. We give a formula for the error committed when the function neither decreases rapidly nor is periodic, so that the sinc series must be truncated for practical purposes. To do so, we first complete a previous result for an odd number of points, before deriving a formula for the more involved case of an even number of points.     

高波数Helmholtz 方程的内罚有限元方法

本文考虑二维和三维区域上高波数Helmholtz 散射问题的线性内罚有限元方法. 该散射问题的边界条件取为一阶吸收边界条件. 本文证明了, 如果加罚参数γ-γ_r+iγ_i 的虚部 γ_i 大于零, 那么内罚有限元方法是绝对稳定的, 即对任意k,h,R > 0 都存在唯一解. 这里k 是波数, h 为网格尺寸, R是区域的直径. 进一步地, 如果|γ_r|≤γ_i≤1, 那么存在与k,h,γ,R 无关的常数C₀;C₁;C₂, 使得当k³h²R ≤ C₀ 时, 该方法的H¹ 误差界为(C₁kh + C₂k³h²R)RM(f, g), 当k³h²R > C₀ 且kh 有界时,H¹ 误差界为(C₁kh + C₂/γ_i)RM(f, g), 其中M(f, g) := (‖f‖_L²(Ω) + R^-1/2‖g‖_L²(Γ)) + R^-1|g|_H^1/2(Γ). 另外, 本文还推导了L² 误差估计. 注意到γ = 0 时内罚有限元方法就是经典的有限元方法, 通过取加罚参数为iγ_>i 并令γ_i 趋于0+, 本文还在k³h²R ≤ C₀ 的条件下, 得到了有限元方法的稳定性和误差估计.作者以前的工作只考虑了加罚参数为纯虚数的情形并且没有考虑对R 的依赖关系.     

CONVERGENCE OF SPECTRAL METHOD IN TIME FOR BURGERS' EQUATION

1.IntroductionTheclassicalspectralmethodsforBurgers'equationUt UUz~AUg.=fusuallydiscretizethetimedirectionwithfinitedifferencemethod[1]sothattheorderofconvergenceinthet-directionislowerthanthatinthex-directionwhichisdiscretisedwithspectralmethod.Therefore,toobtainintegralhighorderofconvergence,wemayalsoapplyspectralmethodtothet-direction.Suchatrialcanbefoundin12],wheresomenumericalresultsweregivenbyusingthetaumethodintime,butconvergenceofthemethodisnotprovedtheoretically.Asweshallfindinpara…     

Stabilized mixed triangular finite elements at large deformations using area bubble functions

In this paper stabilized mixed triangular finite elements are presented in order to avoid volume locking and to damp stress oscillations. Geometrically non-linear elastic problems are addressed. The mixed method of incompatible modes and the mixed method of enhanced strains are considered as special cases. As a key idea, volume and area bubble functions are used for the method of incompatible modes and the enhanced strain method [1], thus giving both the interpretation of a mixed finite element method with stabilization terms. Concerning non-linear problems these are non-linearly dependent on the current deformation state, however, linearly dependent stabilization terms are used [1]. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step is completely avoided. The variational formulation for the standard two-field method, the method of incompatible modes and the enhanced strain method in finite deformation problems is derived for a hyper elastic Neo-Hookean material. In the representative example Cook's membrane problem illustrates the good performance of the presented approaches compared to existing finite element formulations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stresslet resistance functions for low Reynolds number flow using deforming spheres

The stresslets of two rigid spheres in an ambient pure straining flow are obtained at low Reynolds number by defining and solving an equivalent problem of flow around deforming spheres. If the spheres are separated by a small gap, the stresslet of each sphere (the symmetric first moment of the surface stress) is a singular function of the gap width. For spheres in an ambient pure straining flow, the singularities manifest themselves as the slow convergence of numerical calculations. The methods of lubrication theory are used to calculate the singularities in the stresslets and it is shown that these new singularities can be related to singularities already found in other resistance functions. It is also shown that the singular terms can be used to improve the rate of convergence of series expressions for the stresslets. The series expressions then become valid for all separations of the spheres.     

The Reynolds number and large time behaviour for weak solutions of the Navier-Stokes equations

We consider time-dependent perturbationsu of R. Finn's stationary PR-solution of the Navier-Stokes equations, which converges to a constant vectorv as ¦x¦ .We investigate the large time behaviour ofu in the case, that the (energetic) Reynolds numberR _e of has the critical valueR _e =1. We show that perturbationsu with small initial kinetic energy and small driving force become smooth after finite time.     

Aggregation of microglia in 2D with string gradient weighted moving finite elements

Alzheimer's disease (AD) is a severe neurodegenerative disorder characterised by cognitive impairment and dementia. In the AD‐affected brain, microglia cells are up‐regulated and accumulate at senile plaques, the most prominent pathological feature of AD. In order to further study and predict the movement of activated microglia, we utilised their chemotactic properties. Specifically, we formulated the string gradient weighted moving finite element method for a system of partial differential equations in two dimensions, which includes nonlinear diffusion of a different variable found in chemotaxis models. The method was applied successfully to solve highly nonlinear chemorepulsion–chemorepellent models in two dimensions, and the results were compared with one‐dimensional results found previously in the literature. We conclude that the string gradient weighted moving finite element method is easily applied to chemotaxis models, in particular movement and aggregation of microglia, resulting in the ability to study the models extended in two dimensions efficiently. Our study highlights the feasibility and power of mathematical modelling to advance our understanding of pathophysiological processes in neurodegenerative diseases, including AD. Copyright © 2012 John Wiley & Sons, Ltd.     

Solution of instance-based recognition problems with a large number of classes

A learning-based classification problem with a large number of classes is considered. The error-correcting-output-codes (ЕСОС) scheme is optimized. An initial binary matrix is formed at random so that the number of its rows is equal to the number of classes and each column corresponds to the union of several classes in two macroclasses. In the ЕСОС approach, a binary classification problem is solved for every object to be recognized and for every union. The object is assigned to the class with the nearest code row. A generalization of the ЕСОС approach is presented in which a discrete optimization problem is solved to find optimal unions, probabilities of correct classification are used in dichotomy problems, and the degree of dichotomy informativeness is taken into account. If the solution algorithms for the dichotomy problems are correct, the recognition algorithm for the original problem is correct as well.