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June  2015, 35(6): 2659-2677. doi: 10.3934/dcds.2015.35.2659

A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality

Svetlana Matculevich 1, , Pekka Neittaanmäki 2, and  Sergey Repin 2,

1. 

Dept. of Mathematical Information Technology, Faculty of Information Technology, C321.4, Agora, P.O. Box 35, FI-40014, University of Jyväskylä, Finland
2. 

Dept. of Mathematical Information Technology, Faculty of Information Technology, Agora, P.O. Box 35, FI-40014, University of Jyväskylä, Finland, Finland

Received  January 2014 Revised  April 2014 Published  December 2014

We consider evolutionary reaction-diffusion problems with mixed Dirichlet--Robin boundary conditions. For this class of problems, we derive two-sided estimates of the distance between any function in the admissible energy space and the exact solution of the problem. The estimates (majorants and minorants) are explicitly computable and do not contain unknown functions or constants. Moreover, it is proved that the estimates are equivalent to the energy norm of the deviation from the exact solution.
Keywords: a posteriori estimates, Parabolic equations, Poincaré type estimates..
Mathematics Subject Classification: 35K20, 65M15, 65N3.
Citation: Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2659-2677. doi: 10.3934/dcds.2015.35.2659
References:
[1]

G. Acosta and R. G. Durán, An optimal Poincaré inequality in $L^1$ for convex domains, Proc. Amer. Math. Soc., 132 (2004), 195-202. doi: 10.1090/S0002-9939-03-07004-7.

[2]

M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen, 22, (2003), 751-756. doi: 10.4171/ZAA/1170.

[3]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010, xxii+749pp.

[4]

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, in Applied Mathematical Sciences (eds. E.H. Zarantonello and Author 2), Springer-Verlag, New York, 1985, xxx+322pp. doi: 10.1007/978-1-4757-4317-3.

[5]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967.

[6]

R. S. Laugesen and B. A. Siudeja, Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality, J. Differential Equations, 249 (2010), 118-135. doi: 10.1016/j.jde.2010.02.020.

[7]

O. Mali and P. Neittaanmäki and S. Repin, Accuracy Verification Methods. Theory and Algorithms, Computational Methods in Applied Sciences, 32, Springer, Dordrecht, 2014. doi: 10.1007/978-94-007-7581-7.

[8]

S. Matculevich and S. Repin, Computable estimates of the distance to the exact solution of the evolutionary reaction-diffusion equation, Appl. Math. Comput., 247 (2014), 329-347. arXiv:1310.4602. doi: 10.1016/j.amc.2014.08.055.

[9]

P. Neittaanmäki and S. Repin, Reliable methods for computer simulation, error control and a posteriori estimates, in Studies in Mathematics and its Applications, Elsevier, New York, 2004, x+305pp.

[10]

P. Neittaanmäki and P. S. Repin, A posteriori error majorants for approximations of the evolutionary Stokes problem, J. Numer. Math., 18 (2010), 119-134. doi: 10.1515/JNUM.2010.005.

[11]

P. Neittaanmäki and S. Repin, Guaranteed error bounds for conforming approximations of a Maxwell type problem, in Comput. Methods Appl. Sci., Applied and numerical partial differential equations, Springer, New York, (2010), 199-211. doi: 10.1007/978-90-481-3239-3_15.

[12]

L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292. doi: 10.1007/BF00252910.

[13]

S. I. Repin, Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation, Rend. Mat. Acc. Lincei, 13 (2002), 121-133.

[14]

S. Repin, A posteriori estimates for partial differential equations, in Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, (2008), xii+316pp. doi: 10.1515/9783110203042.

[15]

S. Repin and S. Sauter, Functional a posteriori estimates for the reaction-diffusion problem, C. R. Acad. Sci. Paris, 343 (2006), 349-354. doi: 10.1016/j.crma.2006.06.024.

[16]

S. Repin and S. K. Tomar, A posteriori error estimates for approximations of evolutionary convection-diffusion problems, J. Math. Sci. (N. Y.), 170 (2010), 554-566. doi: 10.1007/s10958-010-0100-1.

show all references

References:
[1]

G. Acosta and R. G. Durán, An optimal Poincaré inequality in $L^1$ for convex domains, Proc. Amer. Math. Soc., 132 (2004), 195-202. doi: 10.1090/S0002-9939-03-07004-7.

[2]

M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen, 22, (2003), 751-756. doi: 10.4171/ZAA/1170.

[3]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010, xxii+749pp.

[4]

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, in Applied Mathematical Sciences (eds. E.H. Zarantonello and Author 2), Springer-Verlag, New York, 1985, xxx+322pp. doi: 10.1007/978-1-4757-4317-3.

[5]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967.

[6]

R. S. Laugesen and B. A. Siudeja, Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality, J. Differential Equations, 249 (2010), 118-135. doi: 10.1016/j.jde.2010.02.020.

[7]

O. Mali and P. Neittaanmäki and S. Repin, Accuracy Verification Methods. Theory and Algorithms, Computational Methods in Applied Sciences, 32, Springer, Dordrecht, 2014. doi: 10.1007/978-94-007-7581-7.

[8]

S. Matculevich and S. Repin, Computable estimates of the distance to the exact solution of the evolutionary reaction-diffusion equation, Appl. Math. Comput., 247 (2014), 329-347. arXiv:1310.4602. doi: 10.1016/j.amc.2014.08.055.

[9]

P. Neittaanmäki and S. Repin, Reliable methods for computer simulation, error control and a posteriori estimates, in Studies in Mathematics and its Applications, Elsevier, New York, 2004, x+305pp.

[10]

P. Neittaanmäki and P. S. Repin, A posteriori error majorants for approximations of the evolutionary Stokes problem, J. Numer. Math., 18 (2010), 119-134. doi: 10.1515/JNUM.2010.005.

[11]

P. Neittaanmäki and S. Repin, Guaranteed error bounds for conforming approximations of a Maxwell type problem, in Comput. Methods Appl. Sci., Applied and numerical partial differential equations, Springer, New York, (2010), 199-211. doi: 10.1007/978-90-481-3239-3_15.

[12]

L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292. doi: 10.1007/BF00252910.

[13]

S. I. Repin, Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation, Rend. Mat. Acc. Lincei, 13 (2002), 121-133.

[14]

S. Repin, A posteriori estimates for partial differential equations, in Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, (2008), xii+316pp. doi: 10.1515/9783110203042.

[15]

S. Repin and S. Sauter, Functional a posteriori estimates for the reaction-diffusion problem, C. R. Acad. Sci. Paris, 343 (2006), 349-354. doi: 10.1016/j.crma.2006.06.024.

[16]

S. Repin and S. K. Tomar, A posteriori error estimates for approximations of evolutionary convection-diffusion problems, J. Math. Sci. (N. Y.), 170 (2010), 554-566. doi: 10.1007/s10958-010-0100-1.

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