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Pointwise supercloseness of pentahedral finite elements. (English) Zbl 1204.65135

The authors study the model problem
\[ - \Delta u = f\text{ in }\Omega,\quad u =0\text{ on }\partial \Omega,\tag{1} \]
where \(\Omega = [0,1]^3\) is a rectangular block with boundary \(\partial \Omega\), consisting of faces parallel to the \(x\)-, \(y\)- and \(z\)-axis. The domain is partitioned into subcubes of side \(h\), and each of these is next subdivided into two pentahedra (triangular prisms). Let \(\{\tau^h \}\) denote a uniform family of pentahedral partitions and \(\overline \Omega = \cup_{e \in \tau^h} \overline{e}\), where \(e = D \times L\), \(D\) and \(L\) represent a triangle parallel to \(xy\)-plane and an one-dimensional interval parallel to the \(z\)-axis, respectively. Further a tensor-product polynomial space of degree \(m \times n\) denoted by \(P_{m,n}: q(x,y,z) = \sum_{(i,j,K) \in I_{m,n}} a_{ijK}x^i y^j z^K, a_{ijK} \in R, q \in P_{m,n}\) is introduced. \((P_{m,n} = P_m \otimes P_n\), \(P_m\) and \(P_n\) are the polynomial spaces of degree \(m\) with respect to \((x,y)\) and the polynomial space of degree \(n\) with respect to \(z\), respectively). The indexing set \(I_{m,n}\) satisfies the condition \(I_{m,n}=\{ (i,j,K)\mid i,j,K \geq0, i+j \leq m, K \leq n \}\). The tensor-product interpolation operator \(\Pi^e_{m,n}: H^1_0(e) \rightarrow P_{m,n}(e)\) \((\Pi^e_{m,n}:\Pi^\varepsilon_m \otimes \pi^e_n \otimes \Pi^e_m\), where \(\Pi^e_m\) and \(\pi^e_n\) represent the interpolation operator of degree \(m\) with respect to \((x,y) \in D\) and the interpolation operator of degree \(n\) with respect to \(z \in L\), respectively) is defined. The corresponding weak form of the problem (1) is represented by
\[ a(u,v) = (f,v) \forall v \in H^1_0(\Omega) \]
\((a(u,v) \equiv (\nabla u, \nabla v)_{L_2 (\Omega)} = \int_\Omega \nabla u . \nabla v \;dx\;dy\;dz, (f,v)_{L_2 (\Omega)} = \int_\Omega f\;v\;dx\;dy\;dz)\). The finite element method is to find \(u_h \in S^h_0 (\Omega)(S^h_0 (\Omega) = \{ v \in H^1_0 (\Omega): v|_e \in P_{m,n}e \;\forall e \in \tau ^h\})\) such that: \(a(u_h, v) = (f,v) \;\;\forall v \in S^h_0 (\Omega)\). The interpolation operator \(\Pi_{m,n}\) satisfies the weak estimate:
\[ |a (u-\Pi_{m,n} u,v)| \leq ch^K \|u\|_{t, \infty, \Omega} |v|_{1,1, \Omega}, \]
where \(t = \max \{m+2, n+2\}, K = \min \{m+1, n+1\}, m=1,2\), and \( n \geq 1\). The interpolant of projection type with important characteristics plays a crucial role in the superconvergence analysis for the finite element method. The shape functions are simple (the original functions of the Legendre polynomial systems). Moreover the shape functions possess orthogonality and other good properties. The interpolant of projection type, due to the proposed characteristics has the supercloseness error estimates.
Main result: The authors present for a uniform family of pentahedral partitions of \(\Omega\), and \(u \in W^{1, \infty}, (\Omega) \cap H^1_0(\Omega), [u_h, \Pi_{m,n}u]\) (the pentahedral finite element approximation and the corresponding interpolant to \(u\)) the supercloseness estimate:
\[ |u_h - \Pi_{m,n}u|_{i, \infty, \Omega} \leq ch ^K| \ln h |^\frac{4}{3}\|u\|_{t, \infty, \Omega}, \]
where \( t= \max \{m+2, n+2\}, K = \min \{m+1, n+1\}, m=1,2\) and \( n \geq 1\).

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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References:

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