Negative norm estimates and superconvergence results in Galerkin method for strongly nonlinear parabolic problems. (English) Zbl 1524.65853
Summary: The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree \(r\geq 1\) are used, which improve upon earlier results of O. Axelsson [Numer. Math. 28, 1–14 (1977; Zbl 0341.65067)] requiring for 2d \(r\geq 2\) and for 3d \(r\geq 3\). Based on quasi-projection technique introduced by J. Douglas jun. et al. [Math. Comput. 32, 345–362 (1978; Zbl 0385.65052)], superconvergence result for the error between Galerkin approximation and approximation through quasi-projection is established for the semidiscrete Galerkin scheme. Further, a priori error estimates in Sobolev spaces of negative index are derived. Moreover, in a single space variable, nodal superconvergence results between the true solution and Galerkin approximation are established.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35K58 | Semilinear parabolic equations |
Keywords:
strongly nonlinear parabolic problems; Galerkin method; global optimal error estimate; quasi-projection; negative norm estimate; superconvergenceReferences:
| [1] | Agmon, S., Lectures on Elliptic Boundary Value Problems (1965), Van Nostrand: Van Nostrand Princeton · Zbl 0151.20203 |
| [2] | Arnold, D. N.; Douglas, J., Superconvergence of the Galerkin approximation of a quasilinear parabolic equation in a single space variable, Calcolo, XVI, 345-369 (1979) · Zbl 0435.65094 |
| [3] | Axelsson, O., Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form, Numer. Math., 28, 1-14 (1977) · Zbl 0341.65067 |
| [4] | Bi, C.; Lin, Y., Discontinuous Galerkin method for monotone nonlinear elliptic problems, Int. J. Numer. Anal. Model., 9, 999-1024 (2012) · Zbl 1268.65149 |
| [5] | Bramble, J. H.; Schatz, A. H.; Thomée, V.; Wahlbin, L. B., Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal., 14, 218-241 (1977) · Zbl 0364.65084 |
| [6] | Bramble, J. H.; Schatz, A. H., Higher order local accuracy by averaging in the finite element method, Math. Comput., 31, 94-111 (1977) · Zbl 0353.65064 |
| [7] | Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol. 15 (2008), Springer: Springer New York · Zbl 1135.65042 |
| [8] | Cao, W.; Zhang, Z., Some recent developments in superconvergence of discontinuous Galerkin methods for time-dependent partial differential equations, J. Sci. Comput., 77, 1402-1423 (2018) · Zbl 1407.65154 |
| [9] | Chow, S. S., Finite element error estimates for non-linear elliptic equations of monotone type, Numer. Math., 54, 373-393 (1989) · Zbl 0643.65058 |
| [10] | Douglas, J.; Dupont, T.; Serrin, J., Uniqueness and comparison theorem for nonlinear elliptic equations in divergence form, Arch. Ration. Mech. Anal., 42, 157-168 (1971) · Zbl 0222.35017 |
| [11] | Douglas, J.; Dupont, T.; Wheeler, M. F., A quasi-projection analysis of Galerkin methods of parabolic and hyperbolic equations, Math. Comput., 32, 345-362 (1978) · Zbl 0385.65052 |
| [12] | Feistauer, M.; Zenisek, A., Finite element solution of nonlinear elliptic problems, Numer. Math., 50, 451-475 (1987) · Zbl 0637.65107 |
| [13] | Feistauer, M.; Zenisek, A., Compactness method in the finite element theory of nonlinear elliptic problems, Numer. Math., 52, 147-163 (1988) · Zbl 0642.65075 |
| [14] | Gudi, T.; Nataraj, N.; Pani, A. K., hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems, Numer. Math., 109, 233-268 (2008) · Zbl 1146.65076 |
| [15] | Houston, P.; Robson, J.; Süli, E., Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case, IMA J. Numer. Anal., 25, 726-749 (2005) · Zbl 1084.65116 |
| [16] | Kacur, J., Solution to strongly nonlinear parabolic problems by a linear approximation scheme, IMA J. Numer. Anal., 19, 119-145 (1999) · Zbl 0946.65145 |
| [17] | Kim, M. Y.; Milner, F. A.; Park, E.-J., Some observations on mixed methods for fully nonlinear parabolic problems in divergence form, Appl. Math. Lett., 9, 78-81 (1996) · Zbl 0854.35057 |
| [18] | Krizek, M.; Neittaanmäki, P., On superconvergence techniques, Acta Appl. Math., 9, 175-198 (1987) · Zbl 0624.65107 |
| [19] | (Krizek, M.; Neittaanmäki, P.; Stenberg, R., Finite Element Methods: Superconvergence, Postprocessing, and A Posteriori Estimates. Finite Element Methods: Superconvergence, Postprocessing, and A Posteriori Estimates, Lecture Notes in Pure and Applied Mathematics Series, vol. 196 (1997), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York) · Zbl 0884.00048 |
| [20] | Kundu, S.; Pani, A. K.; Khebchareon, M., On Kirchoff’s model of parabolic type, Numer. Funct. Anal. Optim., 37, 719-752 (2016) · Zbl 1405.65123 |
| [21] | Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear Quasilinear Equation of Parabolic Type (1968), American Math. Soc.: American Math. Soc. Providence, Rhode Island · Zbl 0174.15403 |
| [22] | Jones, L.; Pani, A. K., On superconvergence results and negative norm estimates for a unidimensional single phase Stefan problem, Numer. Funct. Anal. Optim., 16, 153-175 (1995) · Zbl 0834.65126 |
| [23] | Milner, F. A.; Park, E.-J., A mixed finite element method for a strongly nonlinear second-order elliptic problem, Math. Comput., 64, 973-988 (1995) · Zbl 0829.65128 |
| [24] | Pani, A. K.; Sinha, R. K., On superconvergence results and negative norm estimates for parabolic integro-differential equations, J. Integral Equ., 8, 65-98 (1996) · Zbl 0859.65143 |
| [25] | Park, E. J., A primal hybrid finite element method for a strongly nonlinear second-order elliptic problem, Numer. Methods Partial Differ. Equ., 11, 61-75 (1995) · Zbl 0823.65108 |
| [26] | Park, E. J., Mixed finite element methods for generalized Forchheimer flow in porous media, Numer. Methods Partial Differ. Equ., 21, 213-228 (2005) · Zbl 1141.76425 |
| [27] | Sharma, N.; Khebchareon, M.; Pani, A. K., A priori error estimates of expanded mixed FEM for Kirchhoff type parabolic equation, Numer. Algorithms, 83, 125-147 (2020) · Zbl 1434.65274 |
| [28] | Thomée, V., Negative norm estimates and superconvergence in Galerkin methods for parabolic problems, Math. Comput., 34, 93-113 (1980) · Zbl 0454.65077 |
| [29] | Trudinger, N. S., On comparison principle for quasilinear divergence structure equation, Arch. Ration. Mech. Anal., 57, 128-133 (1974) · Zbl 0311.35046 |
| [30] | Wahlbin, L., Superconvergence in Finite Element Methods, Lecture Notes in Mathematics, vol. 1605 (1995), Springer Verlag: Springer Verlag Berlin · Zbl 0826.65092 |
| [31] | Wheeler, M. F., A priori \(L^2\)-error estimates for Galerkin approximation to parabolic partial differential equations, SIAM J. Numer. Anal., 10, 723-758 (1973) · Zbl 0232.35060 |
| [32] | Yadav, S.; Pani, A. K., Superconvergence of a class of expanded discontinuous Galerkin methods for fully nonlinear elliptic problems in divergence form, J. Comput. Appl. Math., 333, 215-234 (2018) · Zbl 1380.65341 |
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