Efficient Higher Order Single Step Methods for Parabolic Problems: Part I

Bramble, James H.; Sammon, Peter H.

doi:10.2307/2006186

Cited by 22 publications

(48 citation statements)

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“…It is well known from the error analysis for elliptic equations that cf., e.g., [6]; thus (4.5) and (4.6) are valid. We further assume, cf.…”

Section: Application To a Quasilinear Equationmentioning

confidence: 95%

Galerkin time-stepping methods for nonlinear parabolic equations

Akrivis¹,

Makridakis²

2004

ESAIM: M2AN

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Abstract.We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.Mathematics Subject Classification. 65M15, 65M50.

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“…It is well known from the error analysis for elliptic equations that cf., e.g., [6]; thus (4.5) and (4.6) are valid. We further assume, cf.…”

Section: Application To a Quasilinear Equationmentioning

confidence: 95%

Galerkin time-stepping methods for nonlinear parabolic equations

Akrivis¹,

Makridakis²

2004

ESAIM: M2AN

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“…In addition, for u E H1 with div u = 0 in ß and for ceH'il W1-00, w E H1 we have that (2)(3)(4)(5)(6)(7)(8)(9) \b(u,v,w)\^ C\\u\\\\v\\xJ\w\\ and if u E H1 with div u = 0 in ß and if v E H1 n L00, w E H1, (2.9') |è(i/,t;,w)|<C||M||||t;||00||w||1.…”

Section: Time Stepping With the Three-step Backward Differentiation Mmentioning

confidence: 99%

“…See Remark 4.4 for the nonconforming case.) S¡¡ will consist of ordered TV-tuples of piecewise polynomials of degree at most r -1 defined on a quasiuniform partition of ß and satisfying, for some constant C independent of h, the approximation property (1.4) inf (||m-xII +h\\u-XWi)<Chs\\u\\s, VuGH'nH'.KKr, and the inverse property (1)(2)(3)(4)(5) llxlli^CA-'llxll, VXGS¿.…”

mentioning

confidence: 99%

“…[6], [4], with a time-independent preconditioning operator leading to the solution of a number of linear systems with the same matrix at each step. In the Navier-Stokes case, due to the "semilinear" nature of the problem "linearization" may be achieved by totally "lagging" the nonlinear term and then solving one linear system per step, corresponding to the time-independent part of the Navier-Stokes operator.…”

mentioning

confidence: 99%

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On a Higher Order Accurate Fully Discrete Galerkin Approximation to the Navier-Stokes Equations

Baker¹,

Dougalis²,

Karakashian³

1982

Mathematics of Computation

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Abstract. We consider approximating the solution of the initial and boundary value problem for the Navier-Stokes equations in bounded two-and three-dimensional domains using a nonstandard Galerkin (finite element) method for the space discretization and the third order accurate, three-step backward differentiation method (coupled with extrapolation for the nonlinear terms) for the time stepping. The resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions; all these linear systems have the same matrix. The resulting approximations of the velocity are shown to have optimal rate of convergence in L2 under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space.

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“…The implementation of these "base" schemes requires solving linear systems of equations with operators that vary from time step to time step. Following Douglas, Dupont and Ewing, [9], and Bramble and Sammon, [7], we shall modify the schemes by using preconditioned iterative methods for the approximate solution of the linear systems and thereby only solve linear systems with the same, time-independent operator at every step. If k is the time step, we show that solving 0(ln(k~x)) systems at each time step suffices to preserve the overall accuracy and stability of the base schemes.…”

mentioning

confidence: 99%

Cosine methods for second-order hyperbolic equations with time-dependent coefficients

Bales¹,

Dougalis²,

Serbin³

1985

Math. Comp.

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Abstract. We analyze efficient, high-order accurate methods for the approximation of the solutions of linear, second-order hyperbolic equations with time-dependent coefficients. The methods are based on Galerkin-type discretizations in space and on a class of fourth-order accurate, two-step, cosine time-stepping schemes. Preconditioned iterative techniques are used to solve linear systems with the same operator at each time step. The schemes are supplemented by single-step high-order starting procedures and need no evaluations of derivatives of operators. L2-optimal error estimates are proved throughout.1. Introduction. In this paper we shall study efficient, high-order accurate methods for the approximation of the solutions of linear, second-order hyperbolic equations with time-dependent coefficients. We shall use Galerkin-type discretizations in the space variables and base the time-stepping scheme on a class of fourth-order accurate, two-step methods generated by rational approximations to the cosine; cf., e.g., [3], [4] for the case of time-independent coefficients. The implementation of these "base" schemes requires solving linear systems of equations with operators that vary from time step to time step. Following Douglas, Dupont and Ewing, [9], and Bramble and Sammon, [7], we shall modify the schemes by using preconditioned iterative methods for the approximate solution of the linear systems and thereby only solve linear systems with the same, time-independent operator at every step. If k is the time step, we show that solving 0(ln(k~x)) systems at each time step suffices to preserve the overall accuracy and stability of the base schemes.Preconditioned iterative techniques have been used for second-order hyperbolic problems already by Ewing in [10], where a nonlinear equation is solved by a second-order accurate, two-step time discretization. In addition, one of us, [5], [6], has used such techniques coupled with up to fourth-order accurate single-step discretizations for the problem (1.1) below, written in first-order system form, cf. [2]. These single-step schemes are based on rational approximations to e'x and give rise to quite different time-stepping methods from the ones that we study here. In this paper we take a different approach, discretizing the second-order equation without reducing it first to a first-order system. Our two-step schemes require then, as

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Efficient Higher Order Single Step Methods for Parabolic Problems: Part I

Abstract: San e e f f i c i e n t , high order methods are discussel for approximating the solution of an initial boundary value problem for a homogeneous parabolic

Cited by 22 publications

References 1 publication

Galerkin time-stepping methods for nonlinear parabolic equations

Galerkin time-stepping methods for nonlinear parabolic equations

On a Higher Order Accurate Fully Discrete Galerkin Approximation to the Navier-Stokes Equations

Cosine methods for second-order hyperbolic equations with time-dependent coefficients

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