Approximation of the first eigenpair of the \(p\)-Laplace operator using web-spline based finite element method. (English) Zbl 1464.74196
Summary: In this article, we consider the eigenvalue problem of the \(p\)-Laplace operator for \(1<p<\infty\) on a bounded domain with homogeneous Dirichlet boundary condition. The weighted extended \(b\)-spline (web-spline) based mesh-free finite element method is used to compute the first eigenpair. This method does not require any mesh generation and therefore, it can be implemented very efficiently. High accuracy can be achieved with relatively low dimensional approximation spaces and less computation time as compared to the usual finite element method. The numerical results are provided for different values of \(p\) on three different types of domains using web-spline of degree one.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 65D07 | Numerical computation using splines |
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
References:
| [1] | Biezuner, R. J.; Brown, J.; Ercole, G.; Martins, E. M., Computing the first eigenpair of the \(p\)-Laplacian via inverse iteration of sublinear supersolutions, J Sci Comput, 52, 1, 180-201 (2012) · Zbl 1255.65205 |
| [2] | Diaz, J. I.; De Thelin, F., On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J Math Anal, 25, 4, 1085-1111 (1994) · Zbl 0808.35066 |
| [4] | Drábek, P.; Huang, Y. X., Bifurcation problems for the \(p\)-Laplacian in \(R^N\), Trans Amer Math Soc, 349, 1, 171-188 (1997) · Zbl 0868.35010 |
| [5] | Montefusco, E.; Rădulescu, V., Nonlinear eigenvalue problems for quasilinear operators on unbounded domains, Nonlinear Diff Eq Appl, 8, 4, 481-497 (2001) · Zbl 1001.35094 |
| [7] | Lindqvist, P., On a nonlinear eigenvalue problem, fall school in analysis lectures, Ber Univ Jyväskylä Math Inst, 68, 33-54 (1995) · Zbl 0838.35094 |
| [8] | García Azorero, J. P.; Peral Alonso, I., Existence and nonuniqueness for the \(p\)-Laplacian, CommUN Partial Diff Eq, 12, 12, 126-202 (1987) · Zbl 0637.35069 |
| [9] | Lindqvist, P., On the equation \(d i v(| \nabla u |^{p - 2} \nabla u) + \lambda | u |^{p - 2} u = 0\), Proc Amer Math Soc, 190, 1, 157-164 (1990) · Zbl 0714.35029 |
| [10] | Adams, R. A., Sobolev spaces (1978), Academic Press: Academic Press Reading, MA |
| [11] | Bognár, G.; Szabó, T., Solving nonlinear eigenvalue problems by using \(p\)-version of FEM, Comput Math Appl, 46, 1, 57-68 (2003) · Zbl 1049.65125 |
| [12] | Horák, J., Numerical investigation of the smallest eigenvalues of the \(p\)-Laplace operator on planar domains, Electron J Diff Eq, 2011, 132, 1-30 (2011) · Zbl 1228.35159 |
| [13] | Lefton, L.; Wei, D., Numerical approximation of the first eigenpair of the \(p\)-Laplacian using finite elements and the penalty method, Numer Funct Anal Optim, 18, 3-4, 389-399 (1997) · Zbl 0884.65103 |
| [14] | Höllig, K.; Reif, U.; Wipper, J., Weighted extended B-spline approximation of Dirichlet problems, SIAM J Numer Anal, 39, 2, 442-462 (2001) · Zbl 0996.65119 |
| [15] | Höllig, K., Finite element methods with B-splines (2003), SIAM: SIAM Philadelphia · Zbl 1020.65085 |
| [16] | Chaudhary, S.; Srivastava, V.; Kumar, V. S.; Srinivasan, B., Web-spline-based mesh-free finite element approximation for \(p\)-Laplacian, Int J Comput Math, 93, 6, 1022-1043 (2016) · Zbl 1355.65152 |
| [17] | Patra, S.; Srinivas Kumar, V., Finite element approiximation using web-splines for the heat equation, Numer Funct Anal Optim, 39, 13, 1423-1439 (2018) · Zbl 1410.41007 |
| [18] | Pfeil, M., Web-approximation elliptischer eigenwertprobleme (2007) · Zbl 1196.65014 |
| [19] | Kawohl, B.; Fridman, V., Isoperimetric estimates for the first eigenvalue of the \(p\)-Laplace operator and the Cheeger constant, Comment Math Univ Carolin, 44, 4, 659-667 (2003) · Zbl 1105.35029 |
| [20] | Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, (Gunning, R. C. (1970), Press, Princeton: Press, Princeton NJ) · Zbl 0212.44903 |
| [21] | Juutinen, P.; Lindqvist, P.; Manfredi, J. J., The ∞-eigenvalue problem, Arch Rational Mech Anal, 148, 2, 89-105 (1999) · Zbl 0947.35104 |
| [23] | Caliari, M.; Zuccher, S., The inverse power method for the \(p(x)\)-Laplacian problem, J Sci Comput, 65, 2, 698-714 (2015) · Zbl 1329.65262 |
| [24] | Sakaguchi, S., Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann Scuola Norm Sup Pisa Cl Sci, 14, 3, 403-421 (1987) · Zbl 0665.35025 |
| [25] | Höllig, K.; Reif, U., Nonuniform web-splines, Comput Aided Geom Des, 20, 5, 277-294 (2003) · Zbl 1069.65502 |
| [26] | Li, X.; Dong, H., Analysis of the element-free Galerkin method for Signorini problems, Appl Math Comput, 346, 41-56 (2019) · Zbl 1428.74210 |
| [27] | Li, X.; Dong, H., The element-free Galerkin method for the nonlinear \(p\)-Laplacian equation, Comput Math Appl, 75, 7, 2549-2560 (2018) · Zbl 1409.65096 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
