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