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Pointwise a posteriori error estimates for monotone semi-linear equations. (English) Zbl 1104.65107
A continuous barrier function \(w\) is determined such that the solution of the continuous problem lies between \(u_h-w\) and \(u_h+w\). Negative norms of the residues are estimated and the continuous maximum principle is applied. The technique that has been used by R. H. Nochetto, K. G. Siebert and A. Veeser [ibid. 95, No. 1, 163–195 (2003; Zbl 1027.65089); SIAM J. Numer. Anal. 42, No. 5, 2118–2135 (2005; Zbl 1095.65099)] to linear problems, is extended here to monotone semi-linear equations. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35J65 Nonlinear boundary value problems for linear elliptic equations
Software:
ALBERTA; ALBERT
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References:
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