Poincaré Type Inequalities for Vector Functions with Zero Mean Normal Traces on the Boundary and Applications to Interpolation Methods
Repin, S. (2019). Poincaré Type Inequalities for Vector Functions with Zero Mean Normal Traces on the Boundary and Applications to Interpolation Methods. In B. N. Chetverushkin, W. Fitzgibbon, Y. Kuznetsov, P. Neittaanmäki, J. Periaux, & O. Pironneau (Eds.), Contributions to Partial Differential Equations and Applications (pp. 411-432). Springer. Computational Methods in Applied Sciences, 47. https://doi.org/10.1007/978-3-319-78325-3_22
Published inComputational Methods in Applied Sciences
© Springer International Publishing AG, part of Springer Nature 2019
We consider inequalities of the Poincaré–Steklov type for subspaces of H1 -functions defined in a bounded domain Ω∈Rd with Lipschitz boundary ∂Ω . For scalar valued functions, the subspaces are defined by zero mean condition on ∂Ω or on a part of ∂Ω having positive d−1 measure. For vector valued functions, zero mean conditions are applied to normal components on plane faces of ∂Ω (or to averaged normal components on curvilinear faces). We find explicit and simply computable bounds of constants in the respective Poincaré type inequalities for domains typically used in finite element methods (triangles, quadrilaterals, tetrahedrons, prisms, pyramids, and domains composed of them). The second part of the paper discusses applications of the estimates to interpolation of scalar and vector valued functions on macrocells and on meshes with non-overlapping and overlapping cells.
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Is part of publicationContributions to Partial Differential Equations and Applications
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