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obstacle problem; a posteriori error estimate; functional majorant; finite element method; variational inequalities; Raviart–Thomas elements
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 one dimensional benchmark introduced by P. Harasim and J. Valdman. Numerical approximation of the solution of the obstacle problem is obtained by the finite element method using bilinear elements on a rectangular mesh. 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, Lagrange multipliers 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 approximation in the energy norm by the difference of energies of discrete and exact solutions and the majorant estimate bounding the difference of energies of discrete and exact solutions by the value of the functional majorant.
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