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Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes
Author(s) -
Adak Dibyendu,
Natarajan E.,
Kumar Sarvesh
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22298
Subject(s) - discretization , mathematics , lipschitz continuity , polygon mesh , convergence (economics) , dimension (graph theory) , backward euler method , a priori and a posteriori , finite element method , scheme (mathematics) , temporal discretization , regular polygon , polynomial , mathematical analysis , geometry , pure mathematics , philosophy , physics , epistemology , economics , thermodynamics , economic growth
In this article, we discuss and analyze new conforming virtual element methods (VEMs) for the approximation of semilinear parabolic problems on convex polygonal meshes in two spatial dimension. The spatial discretization is based on polynomial and suitable nonpolynomial functions, and a Euler backward scheme is employed for time discretization. The discrete formulation of both the proposed schemes—semidiscrete and fully discrete (with time discretization) is discussed in detail, and the unique solvability of the resulted schemes is discussed. A priori error estimates for the proposed schemes (semidiscrete and fully discrete) in H 1 ‐ and L 2 ‐norms are derived under the assumption that the source term f is Lipschitz continuous. Some numerical experiments are conducted to illustrate the performance of the proposed scheme and to confirm the theoretical convergence rates.