Functional a posteriori error estimates for problems with nonlinear boundary conditions. (English) Zbl 1146.65054

Summary: We consider variational inequalities related to problems with nonlinear boundary conditions. We are focused on deriving a posteriori estimates of the difference between exact solutions of such type variational inequalities and any function lying in the admissible functional class of the problem considered. These estimates are obtained by an advanced version of the variational approach earlier used for problems with uniformly convex functionals.
It is shown that the structure of error majorants reflects properties of the exact solution. The majorants provide guaranteed upper bounds of the error for any conforming approximation and possess necessary continuity properties. In the series of numerical tests performed, it was shown that the estimates are explicitly computable, provide sharp bounds of approximation errors, and give high quality indication of the distribution of local (elementwise) errors.


65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
49M25 Discrete approximations in optimal control
Full Text: DOI


[1] DOI: 10.1007/BF01385738 · Zbl 0797.65080
[2] DOI: 10.1002/nme.1620121010 · Zbl 0396.65068
[3] DOI: 10.1137/0715049 · Zbl 0398.65069
[4] DOI: 10.1016/j.cma.2005.06.003 · Zbl 1115.74047
[5] DOI: 10.1016/j.apnum.2004.06.012 · Zbl 1069.65067
[6] DOI: 10.1007/s00211-005-0634-1 · Zbl 1118.65068
[7] DOI: 10.1090/S0025-5718-02-01402-3 · Zbl 0997.65126
[8] DOI: 10.1007/s00211-003-0495-4 · Zbl 1049.65120
[9] DOI: 10.1007/s002110050009 · Zbl 0943.65075
[10] DOI: 10.2307/2005358 · Zbl 0297.65061
[11] DOI: 10.1007/s00211-005-0630-5 · Zbl 1084.65106
[12] DOI: 10.1137/0731016 · Zbl 0806.65064
[13] DOI: 10.1016/0898-1221(96)00030-2 · Zbl 0857.65071
[14] DOI: 10.1090/S0025-5718-99-01190-4 · Zbl 0949.65070
[15] Repin S., Zapiski Nauch. Semin. (POMI) 243 pp 201– (1997)
[16] Repin S., Zapiski Nauch. Semin. (POMI) 271 pp 188– (2000)
[17] Repin S., Zapiski Nauch. Semin. (POMI) 271 pp 188– (2000)
[18] DOI: 10.1137/S0036142900370812 · Zbl 0992.65073
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