Abstract
A new adaptive algorithm is proposed for constructing grids in the hp-version of the finite element method with piecewise polynomial basis functions. This algorithm allows us to find a solution (with local singularities) to the boundary value problem for a one-dimensional reaction-diffusion equation and smooth the grid solution via the adaptive elimination and addition of grid nodes. This algorithm is compared to one proposed earlier that adaptively refines the grid and deletes nodes with the help of an estimate for the local effect of trial addition of new basis functions and the removal of old ones. Results are presented from numerical experiments aimed at assessing the performance of the proposed algorithm on a singularly perturbed model problem with a smooth solution.
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References
O. C. Zienkiewiez, D. W. Kelly, J. P. de Gago, and I. Babuška, “Hierarchical finite element approaches, adaptive refinement and error estimates,” The Mathematics of Finite Elements and Applications (Academic, London, 1982), pp. 313–346.
O. C. Zienkiewiez and A. Craig, “Adaptive refinement, error estimates, multigrid solution, and hierarchical finite element method concepts,” Accuracy Estimates and Adaptive Refinements in Finite Element Computations (Wiley, Chichester, 1986), pp. 25–59.
R. E. Bank and R. K. Smith, “A posteriory error estimates based on hierarchical bases,” SIAM J. Numer. Anal. 30(4), 921–935(1993).
E. S. Nikolaev, “Adaptive mesh refinement in boundary value problems for linear ODE systems,” Vestn. Mosk. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 3, 11–19(2000).
E. S. Nikolaev, “Method of solving a boundary value problem for a second order ODE on a sequence of adaptively refined and coarsened meshes,” Vestn. Mosk. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 4, 5–16 (2004).
E. S. Nikolaev and O. V. Shishkina, “A method of solving the Dirichlet problem for the second-order elliptic equation in a polygonal domain on an adaptively refined mesh sequence,” Vestn. Mosk. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 4, 3–15 (2005).
N. D. Zolotareva and E. S. Nikolaev, “Method for constructing meshes adapting to the solution of boundary value problems for ordinary differential equations of the second and fourth orders,” Differ. Equations. 45(8), 1189–1202(2009).
N. D. Zolotareva and E. S. Nikolaev, “Adaptive p-version of the finite element method for solving boundary value problems for ordinary second-order differential equations,” Differ. Equations. 49(7), 835–848(2013).
N. D. Zolotareva and E. S. Nikolaev, “Stagnation in the p-version of the finite element method,” Moscow Univ. Comput. Math. Cybern. 30(3), 91–99 (2014).
N. D. Zolotareva and E. S. Nikolaev, “Adaptive hp-finite element method for solving boundary value problems for the stationary reaction-diffusion equation,” Comput. Math. Math. Phys. 55(9), 1484–1500 (2015).
N. D. Zolotareva and E. S. Nikolaev, “Implementation and efficiency analysis of an adaptive hp-finite element method for solving boundary value problems for the stationary reactionLLdiffusion equation,” Comput. Math. Math. Phys. 56(5), 764–782 (2016).
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Original Russian Text © N.D. Zolotareva, E.S. Nikolaev, 2016, published in Vestnik Moskovskogo Universiteta, Seriya 15: Vychisiitei'naya Matematika i Kibernetika, 2016, No. 3, pp. 3-15.
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Zolotareva, N.D., Nikolaev, E.S. A comparative analysis of adaptive algorithms in the finite element method for solving the boundary value problem for a stationary reaction-diffusion equation. MoscowUniv.Comput.Math.Cybern. 40, 97–109 (2016). https://doi.org/10.3103/S0278641916030080
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DOI: https://doi.org/10.3103/S0278641916030080