Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems

Cangiani, Andrea; Georgoulis, Emmanuil H.; Sabawi, Younis A.

doi:10.1016/j.cam.2019.112397

Cited by 17 publications

(10 citation statements)

References 31 publications

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“…The stiff part is treated by an implicit scheme, while an explicit scheme treats the second part. The main important point in our preferred method is to reduce the number of iterations and, as a result, cause a decrease in the scheme's computational time; for more detail about numerical methods, see [22][23][24][25][26][27][28]. It can be seen that the approximate solutions of the whole and minimized models are incredibly close.…”

Section: Discussionmentioning

confidence: 99%

Model Reduction and Analysis for ERK Cell Signalling Pathway Using Implicit-Explicit Rung-Kutta Methods

Rasool

Pirdawood

Sabawi

et al. 2022

PJBAS

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Many complex cell signalling pathways and chemical reaction networks include many variables and parameters; this is sometimes a big issue for identifying critical model elements and describing the model dynamics. Therefore, model reduction approaches can be employed as a mathematical tool to reduce the number of elements. In this study, we use a new technique for model reduction: the Lumping of parameters for the simple linear chemical reaction network and the complex cell signalling pathway that is extracellular-signal-regulated kinase (ERK) pathways. Moreover, we propose a high-order and accurate method for solving stiff nonlinear ordinary differential equations. The curtail idea of this scheme is based on splitting the problem into stiff and non-stiff terms. More specifically, stiff discretization uses the implicit method, and nonlinear discretization uses the explicit method. This is consequently leading to a reduction in the computational cost of the scheme.The main aim of this study is to reduce the complex cell signalling pathway models by proposing an accurate numerical approximation Runge-Kutta method. This improves one's understanding of such behaviour of these systems and gives an accurate approximate solution. Based on the suggested technique, the simple model's parameters are minimized from 6 to 3, and the complex models from 11 to 8. Resultsshow that there is a good agreement between the original models and the simplified models.

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Section: Discussionmentioning

confidence: 99%

Model Reduction and Analysis for ERK Cell Signalling Pathway Using Implicit-Explicit Rung-Kutta Methods

Rasool

Pirdawood

Sabawi

et al. 2022

PJBAS

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“…Furthermore, this will give insight about designing adaptive algorithm, which allow use to control the cost of computations. In the future, this Chapter can be extended to the fully discrete case for semilinear parabolic interface problems in L ∞ L 2 ðÞ þ L 2 H 1 ÀÁ and L ∞ L 2 ðÞ norms [18,[20][21][22].…”

Section: Discussionmentioning

confidence: 99%

A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear Parabolic Problems

Sabawi¹

2021

Finite Element Methods and Their Applications

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This Chapter aims to investigate the error estimation of numerical approximation to a class of semilinear parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method and the space discretization uses the finite element method for which the meshes are allowed to change in time. The key idea in our analysis is to adapt the elliptic reconstruction technique, introduced by Makridakis and Nochetto 2003, enabling us to use the a posteriori error estimators derived for elliptic models and to obtain optimal order in L∞H1 for Lipschitz and non-Lipschitz nonlinearities. In this Chapter, some challenges will be addressed to deal with nonlinear term by employing a continuation argument.

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“…Whenever first-order IMEX-RK techniques have generally been applied to deals with the stiff terms of the chemistry implicitly in combustion simulations and hypersonic flow, two IMEX-RK techniques are fourth-order accurate. As a result, there have been several studies that have attracted much interest, and many numerical schemes, like the Euler method, Runge Kutta method, multistep schemes [17-20], Finite difference method [21, [34][35][36][37][38][39][40], and Finite element methods [22][23][24][25], have been proposed over time.…”

Section: The Proposed Methodsmentioning

confidence: 99%

“…The findings of that work indicated that healthcare operations must focus more on the original model parameters. This model can be estimated by using finite element methods see [35][36][37][38][39].…”

Section: Numerical Experimentsmentioning

confidence: 99%

Mathematical Modeling and Analysis for COVID-19 Model by Using Implicit-Explicit Rung-Kutta Methods

et al. 2022

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One of the most common health care problems globally is COVID-19, and also there are an international effort to monitor it have been proposed and discussed. Despite the fact that many studies have been performed based on clinical evidence and confirmed infected cases. However, there is room for additional research since a range of complex criteria are included for later research forecast. As a consequence, mathematical modelling mixed with the numerical simulations is an effective method for estimating main propagation parameters and forecasting disease model dynamics. We study and present some models for the COVID-19 in this paper, which can answer significant questions concerning global health care and implement important notes. The IMEX Runge–Kutta and classical Runge–Kutta methods are two well-known computational schemes to find the solution for such system of differential equations. The results, which are based on these numerical procedures suggested and provide estimated solutions, provide critical answers to this global problem. The amount of recovered, infected, susceptible, and quarantined people in the expectation can be estimated using numerical data. The findings could also aid international efforts to increase prevention and strengthen intervention programs. The findings could also support international efforts to increase prevention and strengthen intervention programs. It is clearly that the proposed methods more accurate and works in a very large interval in time with a few step sizes. That is consequently beginning to a decrease in the computational price of the method. Numerical experiments show that there is a good argument and accurate solutions for solving this type of problem.

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Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems

Cited by 17 publications

References 31 publications

Model Reduction and Analysis for ERK Cell Signalling Pathway Using Implicit-Explicit Rung-Kutta Methods

Model Reduction and Analysis for ERK Cell Signalling Pathway Using Implicit-Explicit Rung-Kutta Methods

A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear Parabolic Problems

Mathematical Modeling and Analysis for COVID-19 Model by Using Implicit-Explicit Rung-Kutta Methods

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