A first order system least squares method for the Helmholtz equation. (English) Zbl 1347.65174
Summary: We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number \(k\), which always leads to a Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual FOSLS problem which is quite different from that of the standard finite element methods, we give an error analysis to the \(hp\)-version of the FOSLS method where the dependence on the mesh size \(h\), the approximation order \(p\), and the wave number \(k\) is given explicitly. In particular, under some assumption of the boundary of the domain, the \(L^2\) norm error estimate of the scalar solution from the FOSLS method is shown to be quasi optimal under the condition that \(kh/p\) is sufficiently small and the polynomial degree \(p\) is at least \(O(\log k)\). Numerical experiments are given to verify the theoretical results.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
Keywords:
first order system least squares method; Helmholtz equation; high wave number; pollution error; stability; error estimateSoftware:
CamelliaReferences:
| [1] | Cai, Z.; Ku, J., The \(L^2\) norm error estimates for the div least-squares method, SIAM J. Numer. Anal., 44, 1721-1734 (2006) · Zbl 1138.76053 |
| [2] | Hetmaniuk, U., Stability estimates for a class of Helmholtz problems, Commun. Math. Sci., 5, 3, 665-678 (2007) · Zbl 1135.35323 |
| [3] | Baskin, D.; Spence, E.; Wunsch, J., Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations · Zbl 1341.35013 |
| [4] | Babuška, I.; Sauter, S. A., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM Rev., 42, 451-484 (2000) · Zbl 0956.65095 |
| [5] | Feng, X.; Wu, H., Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal., 47, 2872-2896 (2009) · Zbl 1197.65181 |
| [6] | Feng, X.; Wu, H., \(h p\)-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 80, 1997-2024 (2011) · Zbl 1228.65222 |
| [7] | Feng, X.; Xing, Y., Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 82, 1269-1296 (2013) · Zbl 1276.65080 |
| [8] | Griesmair, R.; Monk, P., Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Sci. Comput., 49, 291-310 (2011) · Zbl 1246.65211 |
| [9] | Chen, H.; Lu, P.; Xu, X., A hybridizable discontinuous Galerkin method for the Helmholtz equation with high wave number, SIAM J. Numer. Anal., 51, 2166-2188 (2013) · Zbl 1278.65174 |
| [10] | Wu, H., Pre-asymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part I: Linear version, IMA J. Numer. Anal., 34, 1266-1288 (2014) · Zbl 1296.65144 |
| [11] | Zhu, L.; Wu, H., Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: hp version, SIAM J. Numer. Anal., 51, 1828-1852 (2013) · Zbl 1416.65468 |
| [12] | Babuška, I.; Melenk, J. M., The partition of unity method, Internat. J. Numer. Methods Engrg., 40, 727-758 (1997) · Zbl 0949.65117 |
| [13] | Melenk, J.-M.; Babuška, I., The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139, 289-314 (1996) · Zbl 0881.65099 |
| [14] | Cessenat, O.; Despres, B., Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem, SIAM J. Numer. Anal., 35, 255-299 (1998) · Zbl 0955.65081 |
| [15] | Amara, M.; Djellouli, R.; Farhat, C., Convergence analysis of a discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of Helmholtz problems, SIAM J. Numer. Anal., 47, 1038-1066 (2009) · Zbl 1205.65298 |
| [16] | Hiptmair, R.; Moiola, A.; Perugia, I., Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the \(p\)-version, SIAM J. Numer. Anal., 49, 264-284 (2011) · Zbl 1229.65215 |
| [17] | Shen, J.; Wang, L. L., Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains, SIAM J. Numer. Anal., 45, 1954-1978 (2007) · Zbl 1154.65082 |
| [18] | Babuška, I.; Banerjee, U.; Osborn, J., Generalized finite element method - main ideas, results, and perspective, Int. J. Comput. Methods, 1, 67-103 (2004) · Zbl 1081.65107 |
| [19] | Esterhazy, S.; Melenk, J. M., An analysis of discretizations of the Helmholtz equation in \(L^2\) and in negative norms, Comput. Math. Appl., 67830-853, 830-853 (2014) · Zbl 1350.65123 |
| [20] | Melenk, J.-M., On generalized finite element methods (1995), University of Maryland, College Park: University of Maryland, College Park MD, (Ph.D. thesis) |
| [21] | Babuška, I.; Banerjee, U.; Osborn, J., Survey of meshless and generalized finite element methods: A unified approach, Acta Numer., 12, 1-125 (2003) · Zbl 1048.65105 |
| [22] | Engquist, B.; Runborg, O., Computational high frequency wave propagation, Acta Numer., 12, 181-266 (2003) · Zbl 1049.65098 |
| [23] | Chang, C. L., A least-squares finite element method for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 83, 1-7 (1990) · Zbl 0726.65121 |
| [24] | Lee, B.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J., First-order system least-squares for the Helmholtz equation, SIAM J. Sci. Comput., 21, 1927-1949 (2000) · Zbl 0957.65097 |
| [26] | Melenk, J.-M.; Sauter, S., Wave number explicit convergence analysis for Galerkin discretizations of the Helmholtz equations, SIAM J. Numer. Anal., 49, 1210-1243 (2011) · Zbl 1229.35038 |
| [27] | Demkowicz, L., Polynomial exact sequences and projection-based interpolation with application to Maxwell equations, (Lecture Notes in Mathematics (2008), Springer-Verlag) · Zbl 1143.78366 |
| [28] | Melenk, J.-M.; Parsania, A.; Sauter, S., General DG-methods for highly indefinite Helmholtz problems, J. Sci. Comput., 57, 536-581 (2013) · Zbl 1292.65128 |
| [29] | Bouma, T.; Gopalakrishnan, J.; Harb, A., Convergence rates of the DPG method with reduced test space degree, Comput. Math. Appl., 68, 11, 1550-1561 (2014) · Zbl 1364.65239 |
| [30] | Broersen, D.; Stevenson, R., A robust Petrov-Galerkin discretisation of convection-diffusion equations, Comput. Math. Appl., 68, 11, 1605-1618 (2014) · Zbl 1364.65191 |
| [31] | Cai, Z.; Carey, V.; Ku, J.; Park, E. J., Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global \(L^2\) norm, Comput. Math. Appl., 70, 4, 648-659 (2015) · Zbl 1443.65315 |
| [32] | Calo, V. M.; Collier, N. O.; Niemi, A. H., Analysis of the discontinuous Petrov-Galerkin method with optimal test functions for the Reissner-Mindlin plate bending model, Comput. Math. Appl., 66, 12, 2570-2586 (2014) · Zbl 1368.65231 |
| [33] | Carstensen, C.; Gallistl, D.; Hellwig, F.; Weggler, L., Low-order dPG-FEM for an elliptic PDE, Comput. Math. Appl., 68, 11, 503-1512 (2014) · Zbl 1364.65242 |
| [34] | Carstensen, C.; Demkowicz, L.; Gopalakrishnan, J., Breaking spaces and forms for the DPG method and applications including Maxwell equations, Comput. Math. Appl. (2016), in press · Zbl 1359.65249 |
| [35] | Chan, J.; Heuer, N.; Bui-Thanh, T.; Demkowicz, L., A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms, Comput. Math. Appl., 67, 4, 771-795 (2014) · Zbl 1350.65120 |
| [36] | Chan, J.; Evans, J. A.; Qiu, W., A dual Petrov-Galerkin finite element method for the convection-diffusion equation, Comput. Math. Appl., 68, 11, 1513-1529 (2014) · Zbl 1364.65192 |
| [37] | Chen, H.; Fu, G.; Li, J.; Qiu, W., First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems, Comput. Math. Appl., 68, 12, 1635-1652 (2014), part A · Zbl 1369.65144 |
| [38] | Demkowicz, L.; Gopalakrishnan, J., A class of discontinuous Petrov-Galerkin methods. Part I: The transport equation, Comput. Methods Appl. Mech. Engrg., 199, 1558-1572 (2010) · Zbl 1231.76142 |
| [39] | Demkowicz, L.; Gopalakrishnan, J., A class of discontinuous Petrov-Galerkin methods. Part II: Optimal test functions, Numer. Methods Partial Differential Equations, 27, 70-105 (2011) · Zbl 1208.65164 |
| [40] | Demkowicz, L.; Gopalakrishnan, J., A primal DPG method without a first order reformulation, Comput. Math. Appl., 66, 1058-1064 (2013) · Zbl 07863339 |
| [41] | Ellis, T.; Demkowicz, L.; Chan, J., Locally conservative discontinuous Petrov-Galerkin finite elements for fluid problems, Comput. Math. Appl., 68, 11, 1530-1549 (2014) · Zbl 1364.76086 |
| [42] | Gopalakrishnan, J.; Qiu, W., An analysis of the practical DPG method, Math. Comp., 83, 537-552 (2014) · Zbl 1282.65154 |
| [43] | Heuer, N.; Karkulik, M., DPG method with optimal test functions for a transmission problem, Comput. Math. Appl., 70, 5, 1070-1081 (2015) · Zbl 1443.65333 |
| [44] | Heuer, N.; Karkulik, M.; Sayas, F. J., Note on discontinuous trace approximation in the practical DPG method, Comput. Math. Appl., 68, 11, 1562-1568 (2014) · Zbl 1364.65250 |
| [45] | Roberts, N. V.; Bui-Thanh, T.; Demkowicz, L., The DPG method for the stokes problem, Comput. Math. Appl., 67, 4, 966-995 (2014) · Zbl 1381.76200 |
| [46] | Gopalakrishnan, J.; Muga, I.; Olivares, N., Dispersive and dissipative errors in the DPG method with scaled norms for Helmholtz equation, SIAM J. Sci. Comput., 36, A20-A39 (2014) · Zbl 1290.65113 |
| [47] | Roberts, N. V., Camellia: A software framework for discontinuous PetrovGalerkin methods, Comput. Math. Appl., 68, 11, 1581-1604 (2014) · Zbl 1364.65002 |
| [48] | Demkowicz, L., (Computing with \(h p\) Finite Elements. I. Computing with \(h p\) Finite Elements. I, One- and Two-Dimensional Elliptic and Maxwell Problems (2006), Chapman & Hall/CRC Press, Taylor and Francis) |
| [49] | Melenk, J.-M.; Sauter, S., Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79, 1871-1914 (2010) · Zbl 1205.65301 |
| [50] | Demkowicz, L.; Gopalakrishnanb, J.; Mugac, I.; Zitelli, J., Wavenumber explicit analysis of a DPG method for the multidimensional Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 213-216, 126-138 (2012) · Zbl 1243.76059 |
| [51] | Stein, E. M., (Singular Integrals and Differentiability Properties of Functions. Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30 (1970), Princeton University Press: Princeton University Press Princeton, NJ) · Zbl 0207.13501 |
| [52] | Mclean, W., Strongly Elliptic Systems and Boundary Integral Equations (2000), Cambridge University Press, ISBN-10: 052166375X · Zbl 0948.35001 |
| [53] | Melenk, J.-M., (Hp Finite Element Methods for Singular Perturbations. Hp Finite Element Methods for Singular Perturbations, Lecture notes in Mathematics, vol. 1976 (2002), Spring-Verlag), MR1939620 · Zbl 1021.65055 |
| [54] | Demkowicz, L.; Gopalakrishnan, J.; Schöberl, J., Polynomial extension operators. Part III, Math. Comp., 81, 279, 1289-1326 (2012) · Zbl 1252.46022 |
| [55] | Monk, P., On the \(p\)- and \(h p\)-extension of Nédélec’s curl-conforming elements, J. Comput. Appl. Math., 53, 117-137 (1992) · Zbl 0820.65066 |
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.
