## A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations.(English)Zbl 1170.65099

The authors develop some efficient spectral algorithms based on Jacobi-Galerkin methods for the solution of integrated forms of fourth-order differential equations in one and two variables. The spatial approximation is based on Jacobi polynomials $$P_n^{(\alpha ,\beta )}(x)$$, with $$\alpha ,\beta \in (-1,\infty )$$ and $$n$$ the polynomial degree. For $$\alpha =\beta$$ , one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for $$\alpha =\beta =\mp \frac 12,\alpha =\beta =0$$, the Chebyshev polynomials of the first and second kinds and Legendre polynomials, respectively. For the nonsymmetric Jacobi polynomials, the two important special cases $$\alpha =-\beta =\pm \frac 12$$ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two-dimensional version of the approximations is obtained by tensor products of the one-dimensional bases.
The resulting discrete systems have specially structured matrices that can be efficiently inverted. An algebraic preconditioning yields a condition number of $$O(N),$$ ($$N$$ being the the number of retained modes of approximations) which is an improvement with respect to the well-known condition number $$O(N^8)$$ of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to $$N^{d+1}$$ for a $$d$$-dimensional problem. Numerical results are presented in which the usual exponential behaviour of spectral approximations is exhibited.

### MSC:

 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 35J40 Boundary value problems for higher-order elliptic equations 65F35 Numerical computation of matrix norms, conditioning, scaling 65Y20 Complexity and performance of numerical algorithms
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### References:

 [1] Boyd, Chebyshev and Fourier Spectral methods (2001) · Zbl 0994.65128 [2] Coutsias, An efficient spectral method for ordinary differential equations with rational function, Math Comp 65 pp 611– (1996) · Zbl 0846.65037 [3] Funaro, Polynomial approximation of differential equations, Lecturer Notes in Physics (1992) · Zbl 0774.41010 [4] Gottlieb, Numerical analysis of spectral methods: theory and applications (1977) · Zbl 0412.65058 [5] Szegö, Orthogonal polynomials, Am Math Soc Colloq Pub 23 (1985) [6] Canuto, Spectral methods in fluid dynamics (1989) [7] Voigt, Spectral methods for partial differential equations (1984) [8] Phillips, On the coefficients of integrated expansions of ultraspherical polynomials, SIAM J Numer Anal 27 pp 823– (1990) · Zbl 0701.33007 [9] Doha, On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations, J Comput Appl Math 139 pp 275– (2002) · Zbl 0991.33003 [10] Doha, Explicit formulae for the coefficients of integrated expansions for Jacobi polynomials and their integrals, Integral Transforms Special Functions 14 pp 69– (2003) · Zbl 1051.33004 [11] Bialecki, A Legendre spectral collocation method for the biharmonic dirichlet problem, Math Model Numer Anal 34 pp 637– (2000) · Zbl 0984.65121 [12] Bialecki, A Legendre spectral Galerkin method for the biharmonic Dirichlet problem, SIAM J Sci Comput 22 pp 1549– (2000) · Zbl 0986.65115 [13] Bjørstad, Fast numerical solution of the biharmonic Dirichlet problem on rectangles, SIAM J Numer Anal 20 pp 59– (1983) · Zbl 0561.65077 [14] Bjørstad, Efficient algorithms for solving a fourth-order equation with the spectral-Galerkin method, SIAM J Sci Comput 18 pp 621– (1997) · Zbl 0939.65129 [15] Guo, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J Math Anal Appl 243 pp 373– (2000) · Zbl 0951.41006 [16] Guo, Jacobi interpolation approximations and their applications to singular differential equations, Adv Comput Math 14 pp 227– (2001) · Zbl 0984.41004 [17] Babus\~ska, Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two-dimensions, Numer Math 85 pp 219– (2000) [18] Babus\~ska, Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces, I. Approximability of functions in the weighted Besov spaces, SIAM J Numer Anal 39 pp 1512– (2002) [19] Guo, Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, Internat J Numer Methods Eng 53 pp 65– (2002) · Zbl 1001.65129 [20] Stephan, On the convergence of the p-version of the boundary element Galerkin method, Math Comp 52 pp 31– (1989) · Zbl 0661.65118 [21] Wang, A rational approximation and its applications to nonlinear partial differential equations on the whole line, J Math Anal Appl 274 pp 374– (2002) · Zbl 1121.41303 [22] Heinrichs, Improved condition number of spectral methods, Math Comp 53 pp 103– (1989) · Zbl 0676.65115 [23] Shen, Efficient spectral-Galerkin method I: Direct solvers of second-and fourth-order equations using Legendre polynomials, SIAM J Sci Comput 15 pp 1489– (1994) · Zbl 0811.65097 [24] Shen, Efficient spectral-Galerkin method II: Direct solvers of second-and fourth-order equations using Chebyshev polynomials, SIAM J Sci Comput 16 pp 74– (1995) · Zbl 0840.65113 [25] Doha, Efficient spectral-Galerkin algorithms for direct solution of the integrated forms for second-order equations using ultraspherical polynomials, ANZIAM J 48 pp 361– (2007) · Zbl 1138.65104 [26] Fernandino, Jacobi Galerkin spectral method for cylindrical and spherical geometries, Chem Eng Sci 62 pp 6777– (2007) [27] Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, Part I: Derivation of basic equations, Acta Astronaut 4 pp 1177– (1977) · Zbl 0427.76047 [28] Buzbee, The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular domains, SIAM J Numer Anal 8 pp 722– (1971) [29] Heinrichs, Line relaxation for spectral multigrid methods, J Comput Phys 77 pp 166– (1988) · Zbl 0649.65055 [30] Heinrichs, Collocation and full multigrid methods, Appl Math Comput 26 pp 35– (1988) · Zbl 0637.65114 [31] Zang, Spectral multigrid methods for elliptic equations I, J Comput Phys 48 pp 485– (1982) · Zbl 0496.65061 [32] Zang, Spectral mutigrid methods for elliptic equations II, J Comput Phys 54 pp 489– (1984) [33] Bernardi, Some spectral approximations of fourth-order problems (1991) [34] Funaro, Some results about the pseudospectral approximation of one dimensional fourth-order problems, Numer Math 58 pp 399– (1990) · Zbl 0714.65074 [35] Bernardi, Spectral methods for the approximation of fourth-order problems: applications to the stokes and Navier-Stokes equations, Comput Struct 30 pp 205– (1988) · Zbl 0668.76038 [36] C. Bernardi and Y. Maday, Some spectral approximations of one-dimensional fourth-order problems, ICASE, NASA-Langley Research Center, Hampton, VA, 1989. · Zbl 0675.65114 [37] Karageorghis, Efficient direct methods for solving the spectral collocation equations for Stokes flow in rectangularly decomposable domains, SIAM J Sci Statist Comput 10 pp 89– (1989) · Zbl 0665.76039 [38] Karageorghis, Spectral collocation methods for Stokes flow in contraction geometries and unbounded domains, J Comput Phys 80 pp 314– (1989) · Zbl 0659.76037 [39] A. H. Bhrawy, Spectral galerkin method for solving the integrated forms of second- and fourth-order differential equations by using ultraspherical polynomials, Cairo University, Egypt, M.Sc. Thesis, 2003. [40] Luke 2 (1969) [41] Doha, On the coefficients of differentiated expansions and derivatives of Jacobi polynomials, J Phys A Math Gen 35 pp 3467– (2002) · Zbl 0997.33004 [42] Watson, A note on generalized hypergeometric series, Proc London Math Soc 23 (2) pp xiii– (1925) · JFM 51.0283.04 [43] Doha, On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J Phys A Math Gen 37 pp 657– (2004) · Zbl 1055.33007 [44] Doha, The ultrasherical coefficients of the moments of a general-order derivative of an infinitely differentiable function, J Comput Appl Math 89 pp 53– (1998) · Zbl 0909.33007 [45] Luke, Mathematical functions and their approximations (1975) [46] Doha, Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer Algorith 42 pp 137– (2006) · Zbl 1103.65119 [47] Doha, Efficient Spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl Numer Math (2007) · Zbl 1138.65104 [48] Graham, Kronecker products and matrix calculus: with applications (1981) · Zbl 0497.26005 [49] Malek, Pseudospectral collocation methods for fourth order differential equations, IMA J Numer Anal 15 pp 523– (1995) · Zbl 0855.65088 [50] Alperto, A fast algorithm for the evaluation of Legendre expansions, SIAM J Sci Stat Comput 12 pp 158– (1991) [51] Shen, Efficient Chebyshev Legendre Galerkin methods for elliptic problems, Houston J Math pp 233– (1996)
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