Approximation of linear hyperbolic interface problems on finite element: Some new estimates

Adewole, Matthew O.

doi:10.1016/j.amc.2018.12.047

Cited by 6 publications

(6 citation statements)

References 22 publications

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“…Piraux et al in 2001, [12] Suggested a new interface method for the solution of the 1D hyperbolic interface models. Using finite element method (FEM), Adewole approximated linear hyperbolic interface models [13]. The author also explored a numerical algorithm for solving hyperbolic interface models [14].…”

Section: Introductionmentioning

confidence: 99%

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Haar wavelet collocation method for solving hyperbolic type double interfaces problem with discontinuous coefficients

Asif

Farooq

Haider

2023

Preprint

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In this article, we considered wave propagation problems through heterogeneous media or hyperbolic type interface problems. A hybrid numerical technique is presented for the numerical solution (NS) of the these type of problems. The proposed method based on Haar wavelet collocation method (HWCM) and finite difference method (FDM). In this technique, the second order spatial partial derivative is approximated by truncated Haar wavelet series and temporal derivative is approximated by FDM. In case of linear hyperbolic interface problems, the resulting algebraic systems are solved by the Gauss elimination method. While in the case of nonlinear, the nonlinearity of the problem by using quasi-Newton linearization technique. The maximum absolute errors (MAEs), root mean square errors (RMSEs) and computational convergence rate (RcN ) are calculated by utilizing distinct collocation points (CPs). The convergence and stability analysis of the proposed technique are also discussed. Both the theoretical and numerical results affirms that the approximate solution catched the exact solution very well.

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Section: Introductionmentioning

confidence: 99%

“…Using finite element method (FEM), Adewole approximated linear hyperbolic interface models [13]. The author also explored a numerical algorithm for solving hyperbolic interface models [14]. Droubi et al [15] gave a numerical approach to hyperbolic interface models using energy method.…”

Section: Introductionmentioning

confidence: 99%

Haar wavelet collocation method for solving hyperbolic type double interfaces problem with discontinuous coefficients

Asif

Farooq

Haider

2023

Preprint

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“…While in 2018, Ahmedatt et a.l looked at some nonlinear hyperbolic -Laplacian equations [4]. Adewole, in 2019, found the APPS of a linear hyperbolic (LHBVP) [5].…”

Section: Introductionmentioning

confidence: 99%

The Galerkin-Implicit Method for Solving Nonlinear Variable Coefficients Hyperbolic Boundary Value Problem

Al-Hawasy

Mansour

2021

eijs

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This paper has the interest of finding the approximate solution (APPS) of a nonlinear variable coefficients hyperbolic boundary value problem (NOLVCHBVP). The given boundary value problem is written in its discrete weak form (WEFM) and proved have a unique solution, which is obtained via the mixed Galerkin finite element with implicit method that reduces the problem to solve the Galerkin nonlinear algebraic system (GNAS). In this part, the predictor and the corrector techniques (PT and CT, respectively) are proved at first convergence and then are used to transform the obtained GNAS to a linear GLAS . Then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are studied. Some illustrative examples are used, where the results are given by figures that show the efficiency and accuracy for the method.

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“…Finite element solutions of non-interface hyperbolic problems have been extensively discussed in [7,8,9,17,19,22,25]. The convergence of finite element solutions of linear hyperbolic interface problems has been considered in [3,4,14,15,16]. In [16], the authors assumed that the interface can be fitted exactly using interface elements with curved edges and established convergence rates of optimal order for both semi and full discretizations.…”

Section: Introductionmentioning

confidence: 99%

“…With low regularity assumptions on the solution across the interface and with the assumption that the interface could not be fitted exactly, almost optimal convergence rates in L 2 (Ω) and H 1 (Ω) norms were established. In [4], we proposed finite element solution of a linear hyperbolic interface problem where the interface was approximated by straight lines. Quasi-uniform triangular elements were used for the spatial discretization and time discretization was based on a three-step implicit scheme.…”

Section: Introductionmentioning

confidence: 99%

Approximate solution of nonlinear hyperbolic equations with homogeneous jump conditions

Adewole¹

2019

J. Numer. Anal. Approx. Theory

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We present the error analysis of class of second order nonlinear hyperbolic interface problem where the spatial and time discretizations are based on finite element method and linearized backward difference scheme respectively. Both semi discrete and fully discrete schemes are analyzed with the assumption that the interface is arbitrary but smooth. Almost optimal convergence rate in \(H^1(\Omega)\)-norm is obtained. Examples are given to support the theoretical result.

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Approximation of linear hyperbolic interface problems on finite element: Some new estimates

Cited by 6 publications

References 22 publications

Haar wavelet collocation method for solving hyperbolic type double interfaces problem with discontinuous coefficients

Haar wavelet collocation method for solving hyperbolic type double interfaces problem with discontinuous coefficients

The Galerkin-Implicit Method for Solving Nonlinear Variable Coefficients Hyperbolic Boundary Value Problem

Approximate solution of nonlinear hyperbolic equations with homogeneous jump conditions

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