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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Shrnutí: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.
Bibliografie:ObjectType-Article-1
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content type line 14
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-025-02896-z