Verification of functional a posteriori error estimates for obstacle problem in 2D. (English) Zbl 1312.49035

Summary: We verify functional a posteriori error estimates proposed by S. Repin for a class of obstacle problems in two space dimensions. New benchmarks with known analytical solution are constructed based on the one-dimensional benchmark introduced by P. Harasim and J. Valdman. A numerical approximation of the solution of the obstacle problem is obtained by the finite element method using bilinear elements on a rectangular mesh. The error of the approximation is measured by a functional majorant. The majorant value contains three unknown fields: a gradient field discretized by Raviart-Thomas elements, a Lagrange multiplier field discretized by piecewise constant functions and a scalar parameter \(\beta\). The minimization of the majorant value is realized by an alternate minimization algorithm, whose convergence is discussed. Numerical results validate two estimates, the energy estimate bounding the error of the approximation in the energy norm by the difference of the energies of the discrete and exact solutions and the majorant estimate bounding the difference of the energies of the discrete and the exact solutions by the value of the functional majorant.


49M25 Discrete approximations in optimal control
49J40 Variational inequalities
65K15 Numerical methods for variational inequalities and related problems
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
74M15 Contact in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74K05 Strings
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