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Exponential convergence of simplicial hp-FEM for \(H^1\)-functions with isotropic singularities. (English) Zbl 1352.65548
Kirby, Robert M. (ed.) et al., Spectral and high order methods for partial differential equations, ICOSAHOM 2014. Selected papers from the ICOSAHOM conference, June 23–27, 2014, Salt Lake City, UT, USA. Cham: Springer (ISBN 978-3-319-19799-9/hbk; 978-3-319-19800-2/ebook). Lecture Notes in Computational Science and Engineering 106, 435-444 (2015).
Many nonlinear partial differential equations admit solutions which are analytical but exhibit isolated point singularities at a set \(\mathcal{G}\). The hp-version of the finite elements method (hp-FEM) is known to deliver exponential convergence for such problems. In this note the author states an exponential convergence result for a \(C^0\)-conforming hp-FEM on regular, simplicial mesh families with isotropic, geometric refinement towards the singular point(s) \(c\in \mathcal{G}\). These meshes are in addition required to be shape-regular. This type of mesh arises for example in adaptive bisection-tree refinements. Specifically, for singular solutions \(u\in H^1(\Omega)\), where \(\Omega \subset \mathbb R^d\), \(d=2,3\), belonging to a countably normed space with radial weights introduced by M. Costabel et al. [Math. Models Methods Appl. Sci. 22, No. 8, 1250015, 63 p. (2012; Zbl 1257.35056)], the author constructs a continuous, piecewise polynomial interpolant \(I^{hp}u\) which exhibits exponential convergence: there exist constants \(b\), \(C>0\) which depend on \(\Omega\) and on \(u\), in general, such that \[ \| u-I^{hp}u\|_{H^1(\Omega)}\leq C \exp(-bN^{1/(d+1)}). \] Here \(d=2,3\) denotes the space dimension and \(N\) denotes the number of degrees of freedom in the hp-FE approximation. This rate coincides, in space dimensions \(d=1,2,\) with the bounds obtained by W. Gui and I. Babuška [Numer. Math. 49, 577–612 (1986; Zbl 0614.65088); ibid. 49, 613–657 (1986; Zbl 0614.65089); ibid. 49, 659–683 (1986; Zbl 0614.65090)] for corner singularities on structured geometric meshes, and by T. P. Wihler et al. [Comput. Math. Appl. 46, No. 1, 183–205 (2003; Zbl 1059.65095)] on unstructured simplicial geometric meshes. In space dimension \(d=3\), this generalizes the hp-approximation by D. Schötzau et al. [SIAM J. Numer. Anal. 51, No. 4, 2005–2035 (2013; Zbl 1457.65215)] in the case of vertex singularities to instructed, tetrahedral meshes with geometric refinement towards \(\mathcal{G}\).
For the entire collection see [Zbl 06519530].

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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