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Higher order FEM for the obstacle problem of the \(p\)-Laplacian – a variational inequality approach. (English) Zbl 1440.65176

Summary: We consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the \(p\)-Laplacian differential operator for \(p\in(1,\infty)\). We prove an a priori error estimate and convergence rates with respect to the mesh size \(h\) and in the polynomial degree \(q\) under assumed regularity. Moreover, we derive a general a posteriori error estimate which is valid for any uniformly bounded sequence of finite element functions. All our results contain the known results for the linear case of \(p = 2\). We present numerical results on the improved convergence rates of adaptive schemes (mesh size adaptivity with and without polynomial degree adaptation) for the singular case of \(p=1.5\) and for the degenerated case of \(p=3\).

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65K15 Numerical methods for variational inequalities and related problems
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
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