An Adaptive Time Step Procedure for a Parabolic Problem with Blow-up

In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time. We use standard linear elements with mass lumping for the space variable. For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then, we use explicit Runge-Kutta methods for this equivalent problem. In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case. We obtain necessary and sufficient conditions for the blow-up of the numerical solution and prove that the numerical blow-up time converges to the continuous one. We also study, for the explicit Euler approximation, the localization of blow-up points for the numerical scheme. Received October 4, 2001; revised March 27, 2002 Published online: July 8, 2002     

A posteriori error estimates of finite element method for the time-dependent Darcy problem in an axisymmetric domain

We consider the time dependent Darcy problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of these equations in the case of general solution. This discretization relies on a backward Euler’s scheme for the time variable and finite elements for the space variables. We prove a posteriori error estimates that allow for an efficient adaptivity strategy both for the time steps and the meshes. Computations for an example with a known solution are presented which support the a posteriori error estimate.     

Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.     

A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems.     

Numerical approximation of a time-fractional Black–Scholes equation

In this paper a time-fractional Black–Scholes equation is examined. We transform the initial value problem into an equivalent integral–differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization. A rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first-order convergence with respect to the time variable is proved. For overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the Black–Scholes differential operator, we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization. It is proved that the scheme is stable and second-order convergent with respect to the spatial variable. Numerical experiments support these theoretical results.     

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators.     

Nonsmooth data error estimates for damped single step methods for parabolic equations in Banach space

We consider a time discretization method for a parabolic initial boundary value problem obtained from a combination of an A-stable single step method of order p and a lower order method with good smoothing properties. Such methods, including the Crank–Nicolson method combined with the backward Euler method, were analyzed in Hilbert space by Luskin and Rannacher, and nonsmooth data error estimates of order p were obtained. We extend this result to Banach space, and also consider approximations of the time derivative. Further, we apply the results to obtain error estimates in the supremum norm for fully discrete methods obtained by discretizing the space variable by a finite element method. Received: February 1998/ Accepted: November 1998     

Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices

Poroelastic models arise in reservoir modeling and many other important applications. Under certain assumptions, they involve a time-dependent coupled system consisting of Navier–Lamé equations for the displacements, Darcy’s flow equation for the fluid velocity and a divergence constraint equation. Stability for infinite time of the continuous problem and, second and third order accurate, time discretized equations are shown. Methods to handle the lack of regularity at initial times are discussed and illustrated numerically. After discretization, at each time step this leads to a block matrix system in saddle point form. Mixed space discretization methods and a regularization method to stabilize the system and avoid locking in the pressure variable are presented. A certain block matrix preconditioner is shown to cluster the eigenvalues of the preconditioned matrix about the unit value but needs inner iterations for certain matrix blocks. The strong clustering leads to very few outer iterations. Various approaches to construct preconditioners are presented and compared. The sensitivity of the number of outer iterations to the stopping accuracy of the inner iterations is illustrated numerically.     

Pointwise uniformly convergent numerical treatment for the non-stationary Burger–Huxley equation using grid equidistribution

This article presents a numerical method for solving the singularly perturbed Burger–Huxley equation on a rectangular domain. That is, the highest-order derivative term in the equation is multiplied by a very small parameter. This small parameter is known as the perturbation parameter. When the perturbation parameter specifying the problem tends to zero, the solution of the perturbed problem exhibits layer behaviour in the outflow boundary region. Most conventional methods fail to capture this layer behaviour. For this reason, there is much current interest in the development of a robust numerical method that may handle the difficulties occurring due to the presence of the perturbation parameter and the nonlinearity of the problem. To solve both of these difficulties a numerical method is constructed. The first step in this direction is the discretization of the time variable using Euler's implicit method with a constant time step. This produces a nonlinear stationary singularly perturbed semidiscrete problem class. The problem class is then linearized using the quasilinearization process. This is followed by discretization in space, which uses the standard upwind finite difference operator. An extensive amount of analysis is carried out in order to establish the convergence and stability of the proposed method. Numerical experiments are carried out for model problems to illustrate graphically the theoretical results. The results indicate that the scheme faithfully mimics the dynamics of the differential equation.     

Analysis of an extended pressure finite element space for two-phase incompressible flows

We consider a standard model for incompressible two-phase flows in which a localized force at the interface describes the effect of surface tension. If a level set (or VOF) method is applied then the interface, which is implicitly given by the zero level of the level set function, is in general not aligned with the triangulation that is used in the discretization of the flow problem. This non-alignment causes severe difficulties w.r.t. the discretization of the localized surface tension force and the discretization of the flow variables. In cases with large surface tension forces the pressure has a large jump across the interface. In standard finite element spaces, due to the non-alignment, the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure. In many simulations these effects cause large oscillations of the velocity close to the interface, so-called spurious velocities. In [6] it is shown that an extended finite element space (XFEM) is much better suited for the discretization of the pressure variable. In this paper we derive important properties of the XFEM space. We present (optimal) approximation error bounds and prove that the diagonally scaled mass matrix has a uniformly bounded spectral condition number. Results of numerical experiments are presented that illustrate properties of the XFEM space.     

Computation of Self-similar Solution Profiles for the Nonlinear Schrödinger Equation

We discuss the numerical computation of self-similar blow-up solutions of the classical nonlinear Schrödinger equation in three space dimensions. These solutions become unbounded in finite time at a single point at which there is a growing and increasingly narrow peak. The problem of the computation of this self-similar solution profile reduces to a nonlinear, ordinary differential equation on an unbounded domain. We show that a transformation of the independent variable to the interval [0,1] yields a well-posed boundary value problem with an essential singularity. This can be stably solved by polynomial collocation. Moreover, a Matlab solver developed by two of the authors can be applied to solve the problem efficiently and provides a reliable estimate of the global error of the collocation solution. This is possible because the boundary conditions for the transformed problem serve to eliminate undesired, rapidly oscillating solution modes and essentially reduce the problem of the computation of the physical solution of the problem to a boundary value problem with a singularity of the first kind. Furthermore, this last observation implies that our proposed solution approach is theoretically justified for the present problem.     

Primal Discontinuous Galerkin Methods for Time-Dependent Coupled Surface and Subsurface Flow

This paper introduces and analyzes a numerical method based on discontinuous finite element methods for solving the two-dimensional coupled problem of time-dependent incompressible Navier-Stokes equations with the Darcy equations through Beaver-Joseph-Saffman’s condition on the interface. The proposed method employs Crank-Nicolson discretization in time (which requires one step of a first order scheme namely backward Euler) and primal DG method in space. With the correct assumption on the first time step optimal error estimates are obtained that are high order in space and second order in time.     

Finite element analysis of transient heat conduction application of the weighted residual process

The transient heat conduction problem can be solved by application of Galerkin's method to space as well as time discretization. The formulation corresponds to the procedure known as finite elements in time and space. A linear time expansion leads to a step by step technique which is convergent, consistent and absolutely stable. Several numerical examples are presented using two-dimensional isoparametric elements.     

FGMRES preconditioning by symmetric/skew-symmetric decomposition of generalized stokes problems

The generalized Stokes problem is solved for non-standard boundary conditions. This problem arises after time semi-discretization by ALE method of the Navier–Stokes system, which describes the flow of two immiscible fluids with similar densities but different viscosities in a horizontal pipe, when modeling heavy crude oil transportation. We discretized the generalized Stokes problem in space using the “Mini” finite element. The inf-sup condition is proved when the interface between the two fluids and its discretization match exactly. The linear system obtained after discretization is solved using different iterative Krylov methods with and without preconditioning. Numerical experiments with different meshes are presented as well as comparisons between the methods considered. The results suggest that FGMRES and a preconditioning technique based on symmetric/skew-symmetric decomposition is a promising candidate for solving large scale generalized Stokes problem.     

PAC最优的RMAX-KNN探索算法

探索与利用的均衡是强化学习研究的重点之一。探索帮助智能体进一步了解环境来做出更优决策;而利用帮助智能体根据其自身当前对于环境的认知来做出当前最优决策。目前大多数探索算法只与值函数相关联,不考虑当前智能体对于环境的认知程度,探索效率极低。针对此问题,提出了一种基于状态空间自适应离散化的RMAX-KNN强化学习算法,算法根据当前智能体对于环境状态空间的离散化程度改写值函数形式,然后基于此值函数对环境进行合理的探索,逐步实现对于环境状态空间的自适应离散化划分。RMAXKNN算法通过将探索与环境状态空间离散化相结合,逐渐加深智能体对于环境的认知程度,进而提高探索效率,同时在理论上证明该算法是一种概率近似正确(PAC)最优探索算法。在Benchmark环境上的仿真实验结果表明,RMAX-KNN算法可以在探索环境的同时实现对于环境状态空间的自适应离散化,并学习到最优策略。     

Numerical solution of time-fractional fourth-order partial differential equations

A quintic B-spline collocation technique is employed for the numerical solution of time-fractional fourth-order partial differential equations. These equations occur in many applications in real-life problems such as modelling of plates and thin beams, strain gradient elasticity and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical and aerospace engineering. The time-fractional derivative is described in the Caputo sense. Backward Euler scheme is used for time discretization and the quintic B-spline-based numerical method is used for space discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. The given problem is solved with three different boundary conditions, including clamped-type condition, simply supported-type condition, and a transversely supported-type condition. Numerical results are considered to investigate the accuracy and efficiency of the proposed method.     

A fully discrete scheme for the pressure–stress formulation of the time-domain fluid–structure interaction problem

We propose an implicit Newmark method for the time integration of the pressure–stress formulation of a fluid–structure interaction problem. The space Galerkin discretization is based on the Arnold–Falk–Winther mixed finite element method with weak symmetry in the solid and the usual Lagrange finite element method in the acoustic medium. We prove that the resulting fully discrete scheme is well-posed and uniformly stable with respect to the discretization parameters and Poisson ratio, and we provide asymptotic error estimates. Finally, we present numerical tests to confirm the asymptotic error estimates predicted by the theory.     

A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

A finite difference method for a time-dependent singularly perturbed convection–diffusion–reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin–Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin–Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.     

Variable preconditioning in complex Hilbert space and its application to the nonlinear Schrödinger equation

An iterative method is developed for nonlinear equations in complex Hilbert spaces, extending the method of variable preconditioning defined earlier in real spaces. We derive convergence of our method. The motivating example for this extension is the time-dependent nonlinear Schrödinger equation, where we use our iteration for the time discretization of the problem and test it numerically.     

Backward difference formulae for Kuramoto–Sivashinsky type equations

We analyze the discretization of the periodic initial value problem for Kuramoto–Sivashinsky type equations with Burgers nonlinearity by implicit–explicit backward difference formula (BDF) methods, establish stability and derive optimal order error estimates. We also study discretization in space by spectral methods.