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Spectral element methods on unstructured meshes: comparisons and recent advances. (English) Zbl 1102.65119
Summary: Spectral element approximations for triangles are not yet as mature as for quadrilaterals. Here we compare different algorithms and show that using an integration rule based on Gauss-points for simplices is of interest. We point out that this can be handled efficiently and allows to recover the convergence rate theoretically expected, even with curved elements.

MSC:
 65N35 Spectral, collocation and related 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
2Dhp90
Full Text:
References:
 [1] Bos L. (1991). On certain configurations of points in $$\mathbb{R}^n$$ which are unisolvent for polynomial interpolation. J. Approx. Theory 64:271–280 · Zbl 0737.41002 [2] Bos L., Taylor M.A., and Wingate B.A. (2001). Tensor product Gauss-Lobatto points are Fekete points for the cube. Math. Comp. 70:1543–1547 · Zbl 0985.41007 [3] Chen Q., and Babuška I. (1995). Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle. Comput. Methods Appl. Mech. Eng. 128:485–494 [4] Chen Q., and Babuška I. (1996). The optimal symmetrical points for polynomial interpolation of real functions in a tetrahedron. Comput. Methods Appl. Mech. Eng. 137:89–94 · Zbl 0877.65004 [5] Cools R. (2002). Advances in multidimensional integration. J. Comput. Appl. Math. 149:1–12 · Zbl 1013.65019 [6] Demkowicz, L., Walsh, T., Gerdes, K., and Bajer, A. (1998). 2D hp-adaptative finite element package Fortran 90 implementation (2Dhp90), TICAM Report 98–14. [7] Dubiner M. (1991). Spectral methods on triangles and other domains. J. Sci. Comput. 6:345–390 · Zbl 0742.76059 [8] Hesthaven J.S. (1998). From electrostatic to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35:655–676 · Zbl 0933.41004 [9] Hesthaven J.S., and Teng C.H. (2000). Stable spectral methods on tetrahedral elements. SIAM J. Sci. Comput. 21:2352–2380 · Zbl 0959.65112 [10] Hesthaven J.S., and Warburton T. (2002). Nodal high-order methods on unstructured grids. J. Comput. Phys. 181: 186–221 · Zbl 1014.78016 [11] Karniadakis G.E., and Sherwin S.J. (1999). Spectral hp Element Methods for CFD. Oxford University Press, London · Zbl 0954.76001 [12] Pasquetti R., and Rapetti F. (2004). Spectral element methods on triangles and quadrilaterals: comparisons and applications. J. Comput. Phys. 198:349–362 · Zbl 1052.65109 [13] Stroud A.H., and Secrest D. (1966). Gaussian Quadrature Formulas. Prentice Hall, NJ · Zbl 0156.17002 [14] Stroud A.H. (1971). Approximate Calculations of Multiple Integrals. Prentice Hall, NJ · Zbl 0379.65013 [15] Taylor M.A., Wingate B.A., and Vincent R.E. (2000). An algorithm for computing Fekete points in the triangle. SIAM J. Numer. Anal. 38:1707–1720 · Zbl 0986.65017 [16] Taylor M.A., and Wingate B.A. (2000). A generalized diagonal mass matrix spectral element method for non-quadrilateral elements. Appl. Num. Math. 33:259–265 · Zbl 0964.65107 [17] Taylor, M. A., Wingate, B. A., and Bos, L. P. (2004). A new algorithm for computing Gauss-like quadrature points, ICOSAHOM 2004 proc., Brown University. [18] Wandzura S., and Xiao H. (2003). Symmetric quadrature rules on a triangle. Comput. Math. Appl. 45:1829–1840 · Zbl 1050.65022 [19] Warburton T., Pavarino L., and Hesthaven J.S. (2000). A pseudo-spectral scheme for the incompressible Navier–Stokes equations using unstructured nodal elements. J. Comput. Phys. 164:1–21 · Zbl 0961.76063
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