A fourth‐order <i>H</i><sup>1</sup>‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation

Doss, L. Jones Tarcius; Nandini, A. P.

doi:10.1002/num.22306

Cited by 10 publications

(3 citation statements)

References 46 publications

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“…Finite element solutions for the KS equation are not common because the primal variational formulation of fourth-order operators requires finite element basis functions which are piecewise smooth and globally at least C 1 -continuous. Although the KS equation has been studied numerically by several schemes such as local discontinuous Galerkin methods [41], finite elements [11,2], variable mesh finite difference methods [30], B-spline finite difference-collocation method [26], the inverse scattering method [13], a higher-order finite element approach [4], finite difference [1,36,29,31], spectral method [5]. In this paper, a new numeric solution for the KS equation is obtained by introducing a θ-scheme/Adams-Bashforth algorithm for the time discretization and P 1 -type Lagrange polynomials for the spatial approximation.…”

Section: Characterization Of the Robust Controlmentioning

confidence: 99%

Robust Stackelberg Controllability for the Kuramoto-Sivashinsky Equation

Montoya¹,

Breton²

2020

Preprint

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In this article the robust Stackelberg controllability (RSC) problem is studied for a nonlinear fourth-order parabolic equation, namely, the Kuramoto-Sivashinsky equation. When three external sources are acting into the system, the RSC problem consists essentially in combining two subproblems: the first one is a saddle point problem among two sources. Such an sources are called the "follower control" and its associated "disturbance signal". This procedure corresponds to a robust control problem. The second one is a hierarchic control problem (Stackelberg strategy), which involves the third force, so-called leader control. The RSC problem establishes a simultaneous game for these forces in the sense that, the leader control has as objective to verify a controllability property, while the follower control and perturbation solve a robust control problem. In this paper the leader control obeys to the exact controllability to the trajectories. Additionally, iterative algorithms to approximate the robust control problem as well as the robust Stackelberg strategy for the nonlinear Kuramoto-Sivashinsky equation are developed and implemented.

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Section: Characterization Of the Robust Controlmentioning

confidence: 99%

Robust Stackelberg Controllability for the Kuramoto-Sivashinsky Equation

Montoya¹,

Breton²

2020

Preprint

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“…They also established the existence and uniqueness of a continuous solution of fractional KS equation, using the fixed‐point theorem and the Picard–Lindelöf approach. Doss and Nandini 25 adapted H 1 Galerkin finite element method with cubic B‐spline as basis functions. The splitting technique was used, which resulted in a coupled system of equations.…”

Section: Introductionmentioning

confidence: 99%

An improvised extrapolated collocation algorithm for solving Kuramoto–Sivashinsky equation

Shallu

Kukreja

2021

Math Methods in App Sciences

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In the present work, the numerical solution of the Kuramoto–Sivashinsky (KS) equation is obtained using an improvised quintic B‐spline extrapolated collocation technique. This equation helps in the study of wave production in dissipative medium, investigates the hydrodynamic uncertainty in laminar flames, describes the turbulence of diverse physical processes, and so on. In this work, splines are used due to their higher smoothness property and the sparse nature of the matrices corresponding to the B‐splines. The improvised B‐splines are formed by forcing the quintic B‐spline interpolant to satisfy the interpolatory and some special end conditions. These basis functions are used in the collocation method for space integration and a weighted finite difference scheme is used for temporal domain integration. The stability of the technique is analyzed using the von Neumann scheme, and it is found to be unconditionally stable. The proposed method is found to be sixth‐order convergent in space and second‐order convergent in the time direction. The theoretical order of convergence is matching perfectly with the numerical one. The efficiency of this scheme is demonstrated by applying it to a number of examples. The L2, L∞, and global relative errors are calculated and compared with the previous work, especially with those where quintic B‐splines are used as basis functions. Also, the behavior of some KS equations is discussed for which the exact solution is not available. The aim of the paper is to show that such an improvised technique can be implemented to solve partial differential equations like the KS equation.

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“…e nonconforming MFEM brings down the smoothness requirement on FE solution compared to the conforming case. Readers with more interests may refer [11][12][13][14][15] and the references listed. For problem (1), Omrani [1] developed the convergence analysis of the corresponding variables in the semidiscrete and fullydiscrete schemes by using conforming MFEM; however, situation involving nonconforming MFEM was not available till now.…”

Section: Introductionmentioning

confidence: 99%

High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

Wang

Liao

2020

Mathematical Problems in Engineering

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The fourth-order nonlinear Sivashinsky equation is often used to simulate a planar solid-liquid interface for a binary alloy. In this paper, we study the high accuracy analysis of the nonconforming mixed finite element method (MFEM for short) for this equation. Firstly, by use of the special property of the nonconforming EQ1rot element (see Lemma 1), the superclose estimates of order Oh2+Δt in the broken H1-norm for the original variable u and intermediate variable p are deduced for the back-Euler (B-E for short) fully-discrete scheme. Secondly, the global superconvergence results of order Oh2+Δt for the two variables are derived through interpolation postprocessing technique. Finally, a numerical example is provided to illustrate validity and efficiency of our theoretical analysis and method.

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A fourth‐order H¹‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation

Cited by 10 publications

References 46 publications

Robust Stackelberg Controllability for the Kuramoto-Sivashinsky Equation

Robust Stackelberg Controllability for the Kuramoto-Sivashinsky Equation

An improvised extrapolated collocation algorithm for solving Kuramoto–Sivashinsky equation

High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

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A fourth‐order H1‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation

Cited by 10 publications

References 46 publications

Robust Stackelberg Controllability for the Kuramoto-Sivashinsky Equation

Robust Stackelberg Controllability for the Kuramoto-Sivashinsky Equation

An improvised extrapolated collocation algorithm for solving Kuramoto–Sivashinsky equation

High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

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A fourth‐order H¹‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation