hp-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

Kyza, Irene; Metcalfe, Stephen; Wihler, Thomas P.

doi:10.1007/s10915-017-0565-x

Cited by 11 publications

(27 citation statements)

References 26 publications

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“…Estimate (3.17) may be used for the proposition of an efficient adaptive algorithm as in [15,34]. This estimate is more appropriate for adaptivity due its local nature.…”

Section: A Posteriori Error Bound In the Lmentioning

confidence: 99%

“…If this term is large, again the estimates will be unpractical for uniform partitions. However, as in the case of [15,34], the proposition of an efficient time-space adaptive algorithm (using the local estimate (3.20)) could provide a way to control the error.…”

Section: A Posteriori Error Bound In the Lmentioning

confidence: 99%

“…[15,32]. In fact, as illustrated in [15,34], this term enables the proposition of an efficient time-space adaptive algorithm leading to blowup detection and accurate numerical approximation of blowup times. In the cases of NLS equations (1.1) we expect that the exponential term will also be proven beneficial towards the development of an efficient time-space adaptive algorithm in the spirit of [15].…”

Section: Introductionmentioning

confidence: 99%

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A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

Katsaounis¹,

Kyza²

2018

SIAM J. Numer. Anal.

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Abstract. We provide a posteriori error estimates in the L ∞ ([0, T ]; L 2 (Ω))−norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schrödinger equations up to the critical exponent. In particular for the discretization in time we use the relaxation Crank-Nicolson-type scheme introduced by Besse in [9]. The space discretization consists of finite element spaces that are allowed to change between time steps. The estimates are obtained using the reconstruction technique. Through this technique the problem is converted to a perturbation of the original partial differential equation and this makes it possible to use nonlinear stability arguments as in the continuous problem. Our analysis includes as special cases the cubic and quintic nonlinear Schrödinger equations in one spatial dimension and the cubic nonlinear Schrödinger equation in two spatial dimensions. Numerical results illustrate that the estimates are indeed of optimal order of convergence.

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“…Estimate (3.17) may be used for the proposition of an efficient adaptive algorithm as in [15,34]. This estimate is more appropriate for adaptivity due its local nature.…”

Section: A Posteriori Error Bound In the Lmentioning

confidence: 99%

Section: A Posteriori Error Bound In the Lmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

Katsaounis¹,

Kyza²

2018

SIAM J. Numer. Anal.

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“…A posteriori error analysis for stationary and evolution problems has been widely investigated over the last 30 years or so, and significant developments have been achieved, see, e.g., the treatises [1,51] and the references therein for elliptic problems, the works [11,[17][18][19]22,27,29,35,36,38,40,42,43,50] for space-discrete or fully-discrete parabolic problems, or [2,[4][5][6]26,34,41,46] for a posteriori error bounds for time semi-discretisations of evolution PDEs; the above lists contain the results most relevant to this work from the growing literature in the area and are, of course, far from constituting a complete bibliographical account.…”

Section: Introductionmentioning

confidence: 99%

“…This is in contrast with the increasing number of interesting works on a posteriori error analyses for low order time-stepping schemes for nonlinear evolution problems; see, e.g., [8,10,12,17,36,52] and the references therein. At the same time, there exist only few works on the a posteriori error analysis of high order time-stepping schemes for time-discrete nonlinear parabolic problems [30,33,34,41,46].…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Analysis for Implicit–Explicit hp-Discontinuous Galerkin Timestepping Methods for Semilinear Parabolic Problems

2020

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A posteriori error estimates in the L ∞ (H)and L 2 (V)-norms are derived for fully-discrete space-time methods discretising semilinear parabolic problems; here V → H → V * denotes a Gelfand triple for an evolution partial differential equation problem. In particular, an implicit-explicit variable order (hp-version) discontinuous Galerkin timestepping scheme is employed, in conjunction with conforming finite element discretisation in space. The nonlinear reaction is treated explicitly, while the linear spatial operator is treated implicitly, allowing for time-marching without the need to solve a nonlinear system per timestep. The main tool in obtaining these error estimates is a recent space-time reconstruction proposed in Georgoulis et al. (A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems, Submitted for publication) for linear parabolic problems, which is now extended to semilinear problems via a non-standard continuation argument. Some numerical investigations are also included highlighting the optimality of the proposed a posteriori bounds. Keywords A posteriori error analysis • Semilinear PDEs • hp-discontinuous Galerkin timestepping • Space-time finite element methods • Reconstructions • Implicit-explicit timestepping methods B Emmanuil H.

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Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up

Holm

Wihler

2017

Numer. Math.

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We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to nonlinear initial value problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.This effect is commonly termed (finite-time) blow-up.Galerkin Time Stepping. Galerkin-type time stepping methods for initial-value problems are based on weak formulations. For both the cG and the dG time stepping schemes, the test spaces consist of polynomials that are discontinuous at the time nodes. In this way, the discrete Galerkin formulations decouple into local problems on each time step, and the discretizations can therefore be understood as implicit one-step schemes. Galerkin time stepping methods have been analyzed for ordinary differential equations (ODEs), e.g., in [3,[5][6][7][8]10].A key feature of Galerkin time stepping methods is their great flexibility with respect to the size of the time steps and the local approximation orders, thereby naturally leading to an hp-version Galerkin framework. The hp-versions of the cG and dG time stepping schemes were introduced and analyzed in the works [12,13,15,19]. In particular, in the articles [12,19], which focus on ordinary initial value problems with uniform Lipschitz nonlinearities, the use of the contraction mapping theorem made it possible to prove existence and uniqueness results for discrete Galerkin solutions, which are independent of the local approximation orders. We emphasize that the hp-approach is well-known for its ability to approximate smooth solutions with possible local singularities at high algebraic or even exponential rates of convergence; see, e.g., [4,13,14,18] for the numerical approximation of problems with start-up singularities.

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hp-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

Cited by 11 publications

References 26 publications

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

A Posteriori Error Analysis for Implicit–Explicit hp-Discontinuous Galerkin Timestepping Methods for Semilinear Parabolic Problems

Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up

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