Abstract
We study the behavior of orthogonal polynomials on triangles and their coefficients in the context of spectral approximations of partial differential equations. In these spectral approximations one studies series expansions \(u=\sum _{k=0}^{\infty } \hat{u}_k \phi _k\) where the \(\phi _k\) are orthogonal polynomials. Our results show that for any function \(u\in C^{\infty }\) the series expansion converges faster than with polynomial order.
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Acknowledgments
The authors would like to thank two anonymous referees whose critical remarks helped to bring the paper into a better shape. The work reported on in this paper was made possible by the DFG via the project SO 363/11-1. The second author would like to express his gratitude to the DFG.
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Appendix
Appendix
First of all we repeat some of the properties of the Jacobi polynomials which we need in this paper. For further properties see also [16]. The Jacobi polynomials \(P_n^{\alpha ,\beta }(x)\) satisfy following properties:
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\(P_n^{\alpha ,\beta }(x)\) are orthogonal on \([-1,1]\) with respect to the weight function \(w(x)=(1-x)^\alpha (1+x)^\beta , \alpha >-1\) and \(\beta >-1\). It holds
$$\begin{aligned} \int \limits _{-1}^1w(x)P_n^{\alpha ,\beta }(x)P_k^{\alpha ,\beta }(x)dx=\frac{\delta _{n,k}2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}\frac{\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{n!\Gamma (n+\alpha +\beta +1)}. \end{aligned}$$ -
\(P_n^{\alpha ,\beta }(1)=\small \left(\begin{array}{c}{n+\alpha }\\ {n}\end{array}\right)\)
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\(P_n^{\alpha ,\beta }(x)=(-1)^n P_n^{\beta ,\alpha }(-x)\)
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If \(q=\max \{\alpha ,\beta \}\ge -\frac{1}{2}\), then \(\max \nolimits _{-1\le x\le 1}|P_n^{\alpha ,\beta }(x)|=\small \left(\begin{array}{c}{n+q}\\ {n}\end{array}\right)\) holds.
Now we prove the estimates of the orthogonal polynomials which are stated in Sect. 2.
Proof
(of Lemma 2.2)
With the inner product
it follows
\(\square \)
In the proof of Lemma 2.3 we prove only estimate (4) and equation (8). The inequality (5) follows directly from the fact that Jacobi polynomials take their maximum at the boundary. The proof of (6) and (7) is a combination of the approach taken in in (4) and (5).
Proof
(of Lemma 2.3)
In the proof we use the estimate
with \(\alpha , \beta \ge 0, n\in \mathbb{N }_0, g_n^{\alpha ,\beta }(x)\) defined by
and \(C\) is a positive constant less than 12, see [5, p.1]. With this inequality and the relation
we estimate the factors of the APK polynomials in \({\stackrel{\circ }{\mathbb{T }}}\). We have
and
Hence we get an upper bound for all points \((x,y)\in {\stackrel{\circ }{\mathbb{T }}}\) as
with
Finally we have to demonstrate equation (8). To this end we examine the factor
The series expansion of the Jacobi polynomials yields
If \(l\ne 0\) and \((x,y)=(1,0)\) then the value of the series is \(0\) and so part I is proven. If \(l=0\) we get
\(\square \)
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Öffner, P., Sonar, T. Spectral convergence for orthogonal polynomials on triangles. Numer. Math. 124, 701–721 (2013). https://doi.org/10.1007/s00211-013-0530-z
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DOI: https://doi.org/10.1007/s00211-013-0530-z