Semi and Fully Discrete Analysis of Extended Fisher–Kolmogorov Equation with Nonstandard FEMs for Space Discretisation

This article discusses lowest-order nonstandard finite element methods for space discretisation and backward Euler scheme for time discretisation of the extended Fisher–Kolmogorov equation with clamped boundary conditions. Spatial discretisation employs popular piecewise quadratic schemes based on t...

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Vydáno v:	Journal of scientific computing Ročník 104; číslo 1; s. 14
Hlavní autoři:	Das, Avijit, Nataraj, Neela, Chirappurathu Remesan, Gopikrishnan
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.07.2025 Springer Nature B.V
Témata:	Algorithms Approximation Boundary conditions Computational Mathematics and Numerical Analysis Discretization Equilibrium Error analysis Estimates Exact solutions Finite element method Galerkin method Hypotheses Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Norms Phase transitions Propagation Theoretical Ritz projection 35J40 (Secondary) 65N15 Time-dependent Error estimates Extended Fisher–Kolmogorov equation Fourth–order parabolic equation Stability Nonstandard finite element methods 35J35 (Primary) 65N30
ISSN:	0885-7474, 1573-7691
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Abstract	This article discusses lowest-order nonstandard finite element methods for space discretisation and backward Euler scheme for time discretisation of the extended Fisher–Kolmogorov equation with clamped boundary conditions. Spatial discretisation employs popular piecewise quadratic schemes based on triangles, namely, the Morley, the discontinuous Galerkin, and the C 0 interior penalty schemes. Based on the smoother J I M defined for a piecewise smooth input function by a (generalized) Morley interpolation I M followed by a companion operator J from Carstensen and Nataraj (ESAIM Math Model Numer Anal 56(1):41–78, 2022), a smoother based Ritz projection operator is defined. A set of abstract hypotheses establish the approximation properties of the Ritz projection operator. The approach allows for an elegant semidiscrete and fully discrete error analysis with minimal regularity assumption on the exact solution. Error estimates for both the semidiscrete and fully discrete schemes are presented. The numerical results validate the theoretical estimates and demonstrate the capability of the discontinuous Galerkin method to approximate the solution, even for non-smooth initial condition.
AbstractList	This article discusses lowest-order nonstandard finite element methods for space discretisation and backward Euler scheme for time discretisation of the extended Fisher–Kolmogorov equation with clamped boundary conditions. Spatial discretisation employs popular piecewise quadratic schemes based on triangles, namely, the Morley, the discontinuous Galerkin, and the C0 interior penalty schemes. Based on the smoother JIM defined for a piecewise smooth input function by a (generalized) Morley interpolation IM followed by a companion operator J from Carstensen and Nataraj (ESAIM Math Model Numer Anal 56(1):41–78, 2022), a smoother based Ritz projection operator is defined. A set of abstract hypotheses establish the approximation properties of the Ritz projection operator. The approach allows for an elegant semidiscrete and fully discrete error analysis with minimal regularity assumption on the exact solution. Error estimates for both the semidiscrete and fully discrete schemes are presented. The numerical results validate the theoretical estimates and demonstrate the capability of the discontinuous Galerkin method to approximate the solution, even for non-smooth initial condition. This article discusses lowest-order nonstandard finite element methods for space discretisation and backward Euler scheme for time discretisation of the extended Fisher–Kolmogorov equation with clamped boundary conditions. Spatial discretisation employs popular piecewise quadratic schemes based on triangles, namely, the Morley, the discontinuous Galerkin, and the C 0 interior penalty schemes. Based on the smoother J I M defined for a piecewise smooth input function by a (generalized) Morley interpolation I M followed by a companion operator J from Carstensen and Nataraj (ESAIM Math Model Numer Anal 56(1):41–78, 2022), a smoother based Ritz projection operator is defined. A set of abstract hypotheses establish the approximation properties of the Ritz projection operator. The approach allows for an elegant semidiscrete and fully discrete error analysis with minimal regularity assumption on the exact solution. Error estimates for both the semidiscrete and fully discrete schemes are presented. The numerical results validate the theoretical estimates and demonstrate the capability of the discontinuous Galerkin method to approximate the solution, even for non-smooth initial condition.
ArticleNumber	14
Author	Das, Avijit Chirappurathu Remesan, Gopikrishnan Nataraj, Neela
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Cites_doi	10.1016/j.camwa.2021.08.010 10.1016/j.cam.2012.12.019 10.1093/imanum/dru054 10.1016/j.camwa.2024.04.013 10.4208/jcm.1908-m2018-0174 10.1090/S0025-5718-00-01230-8 10.1016/0001-8708(78)90130-5 10.1137/S0036141095280955 10.1137/17M1116362 10.1051/m2an/2021085 10.1103/PhysRevLett.49.1332 10.1007/978-0-387-70914-7 10.1007/s00211-023-01356-w 10.1103/PhysRevLett.60.2641 10.1103/PhysRevLett.35.1678 10.1051/m2an/2014062
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SubjectTerms	Algorithms Approximation Boundary conditions Computational Mathematics and Numerical Analysis Discretization Equilibrium Error analysis Estimates Exact solutions Finite element method Galerkin method Hypotheses Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Norms Phase transitions Propagation Theoretical
Title	Semi and Fully Discrete Analysis of Extended Fisher–Kolmogorov Equation with Nonstandard FEMs for Space Discretisation
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