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Adaptation of the composite finite element framework for semilinear parabolic problems

Authors

  • Anjaly Anand National Institute of Technology, Calicut, India
  • Tamal Pramanick National Institute of Technology, Calicut, India https://orcid.org/0000-0001-8577-0431

DOI:

https://doi.org/10.33993/jnaat531-1392

Keywords:

approximation errors, semilinear parabolic equation, Finite Element Method, convergence
Abstract views: 59

Abstract

In this article, we discuss one type of finite element method (FEM), known as the composite finite element method (CFE). Dimensionality reduction is the primary benefit of CFE as it helps to reduce the complexity of the domain space. The number of degrees of freedom is greater in standard FEM compared to CFE. We consider the semilinear parabolic problem in a 2D convex polygonal domain. The analysis of the semidiscrete method for the problem is carried out initially in the CFE framework. Here, the discretization is carried out only in space. Then, the fully discrete problem is taken into account, where both the spatial and time components get discretized.
In the fully discrete case, the backward Euler method and the Crank-Nicolson method in the CFE framework are adapted for the semilinear problem. The properties of convergence are derived and the error estimates are examined. It is verified that the order of convergence is preserved. The results obtained from the numerical computations are also provided.

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References

R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

R.A. Adams and J.J. Fournier, Sobolev Spaces, Academic Press, 2003.

H. Amann, Existence and stability of solutions for semi-linear parabolic systems, and applications to some diffusion reaction equations, Proc. Roy. Soc. Edinburgh Sect. A, 81 (1978), pp. 35-47. https://doi.org/10.1017/S0308210500010428

A.K. Aziz and P. Monk, Continuous finite elements in space and time for the heat equation, Math. Comp., 52 (1989), pp. 255-274. https://doi.org/10.1090/S0025-5718-1989-0983310-2

J. Becker, A second order backward difference method with variable steps for a parabolic problem, BIT, 38 (2002), pp. 644-664. https://doi.org/10.1007/BF02510406

J.H. Bramble, Discrete methods for parabolic equations with time-dependent coefficients, Numer. Methods for PDE’s, Academic Press, 1979, pp. 41-52. https://doi.org/10.1016/B978-0-12-546050-7.50007-1

J.H. Bramble and P. Sammon, Efficient higher order single step methods for parabolic problems: Part I, Math. Comp., 35 (1980), pp. 655-677. https://doi.org/10.1090/S0025-5718-1980-0572848-X

P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Class. Appl. Math., Society for Industrial and Applied Mathematics, Philadelphia, 2002. https://doi.org/10.1137/1.9780898719208

J. Douglas, Jr. and T.F. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (1970), pp. 575-626. https://doi.org/10.1137/0707048

K. Eriksson, C. Johnson and V. Thomee´ , Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Mod´el. Math. Anal. Num´er., 19 (1985), pp. 611-643. https://doi.org/10.1051/m2an/1985190406111

D.J. Estep, M.G. Larson and R.D. Williams, Estimating the error of numerical solutions of reaction-diffusion equations, American Mathematical Soc., Vol. 146, No. 696, 2000. https://doi.org/10.1090/memo/0696

W. Hackbusch and S.A. Sauter, Adaptive composite finite elements for the solution of PDEs containing nonuniformely distributed micro-scales, Mater. Model., 8:9 (1996), pp. 31-43.

W. Hackbusch and S.A. Sauter, Composite finite elements for problems containing small geometric details. Part II: Implementation and numerical results, Comput. Vis. Sci., 1:1 (1997), pp. 15-25. https://doi.org/10.1007/s007910050002

W. Hackbusch and S.A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math., 75:4 (1997), pp. 447-472. https://doi.org/10.1007/s002110050248

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, 2006.

H.-P. Helfrich, Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data, Numer. Math., 51:5 (1987), pp. 559-569. https://doi.org/10.1007/BF01400356

P. Henry-Labordere, N. Oudjane, X. Tan, N. Touzi and X. Warin, Branching diffusion representation of semilinear PDEs and Monte Carlo approximation, Annales de l’Institut Henri Poincar´e, Probabilit´es et Statistiques, 55:1 (2019), pp. 184-210. https://doi.org/10.1214/17-AIHP880

M. Hutzenthaler, A. Jentzen, T. Kruse, T.A. Nguyen and P.V. Wurstemberger, Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, Proc. R. Soc. A, Math. Phys. Eng. Sci., 476:2244 (2020), pp. 20190630. https://doi.org/10.1098/rspa.2019.0630

M. Hutzenthaler, A. Jentzen and T. Kruse, Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, Partial Differential Eq. Appl., 2:6 (2021), pp. 1-31. https://doi.org/10.1007/s42985-021-00089-5

S. Larsson, Semilinear parabolic partial differential equations: thoery, approximation and applications, New Trends in the Mathematical and Computer Sciences, Vol. 3 (2006), pp. 153-194.

F. Liehr, T. Preusser, M. Rumpf, S. Sauter and L.O. Schwen, Composite finite elements for 3D image based computing, Comput. Visualization Sci. 12:4 (2009), pp. 171-188. https://doi.org/10.1007/s00791-008-0093-1

T. Pramanick and R.K. Sinha, Two-scale composite finite element method for parabolic problems with smooth and nonsmooth initial data, J. Appl. Math. Comput., 58:1-2 (2018), pp. 469-501. https://doi.org/10.1007/s12190-017-1153-9

T. Pramanick and R.K. Sinha, Error estimates for two-scale composite finite element approximations of parabolic equations with measure data in time for convex and nonconvex polygonal domains, Appl. Numer. Math., 143 (2019), pp. 112-132. https://doi.org/10.1016/j.apnum.2019.03.009

T. Pramanick and R.K. Sinha, Composite finite element approximation for parabolic problems in nonconvex polygonal domains, Comp. Methods Appl. Math., 20:2 (2020), pp. 361-378. https://doi.org/10.1515/cmam-2018-0155

M. Rech, Composite finite elements: An adaptive two-scale approach to the nonconforming approximation of Dirichlet problems on complicated domains, PhD thesis, Universit¨at Z¨urich, 2006.

M. Rech, S.A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math., 102:4 (2006), pp. 681-708. https://doi.org/10.1007/s00211-005-0654-x

S. Sarraf, E.J. Lopez, V.E. Sonzogni and M. B. Bergallo ´ , An Algebraic Composite Finite Element Mesh Method, Cuadernos de Matem´atica y Mec´anica, (2009).

S.A. Sauter and R. Warnke, Composite finite elements for elliptic boundary value problems with discontinuous coefficients, Computing, 77:1 (2006), pp. 29-55. https://doi.org/10.1007/s00607-005-0150-2

V. Thomee´ , Galerkin Finite Element Methods for Parabolic Problems (Second Edition), Springer Ser. Comput. Math., Springer-Verlag, Berlin, 2006.

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Ambrosius Barth, Heidelberg, 1995.

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Published

2024-04-16

How to Cite

Anand, A., & Pramanick, T. (2024). Adaptation of the composite finite element framework for semilinear parabolic problems. J. Numer. Anal. Approx. Theory. https://doi.org/10.33993/jnaat531-1392

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