Nonconforming spectral element method: a friendly introduction in one dimension and a short review in higher dimensions

• Adjerid S, Lin T (2007) Higher order immersed discontinuous Galerkin methods. Int J Inf Syst Sci 3(4):555–568

  MathSciNet  MATH  Google Scholar 

• Adjerid S, Lin T (2009) A \(p^{th}\) degree immersed finite element method for boundary value problems. Appl Numer Math 59:1303–1321

  MathSciNet  MATH  Google Scholar 

• Aitbayev R (2013) Existence and uniqueness for a two point interface boundary value problem. Electron J Diff Eqs 242:1–12

  MathSciNet  MATH  Google Scholar 

• Aitbayev R, Yergali Yeva N (2014) A fourth order collocation scheme for two point interface boundary value problems, Adv Numer Anal 2014:875013. https://doi.org/10.1155/2014/875013

• Albright J, Epshteyn Y, Steffen KR (2015) High-order accurate difference potentials methods for parabolic problems. Appl Numer Math 93:87–106

  MathSciNet  MATH  Google Scholar 

• Attanayake C, Chou SH, Deng Q (2022) Higher order SGFEM for one dimensional interface elliptic problems with discontinuous solutions, arXiv:2204.07665v1

• Axelsson O, Barker VA (2001) Finite element solution of boundary value problems, theory and computation. SIAM

• Babuska I (1970) The finite element method for elliptic equations with discontinuous coefficients. Computing 5:207–213

• Babuska I, Elman HC (1989) Some aspects of parallel implementation of the finite element method on message passing architectures. J Comput Appl Math 27:157–187

  MathSciNet  MATH  Google Scholar 

• Babuska I, Guo B (1988) On regularity of the solutions of elliptic problems with piecewise analytic data, part I: boundary value problems or linear elliptic equation of second order. SIAM J Math Anal 19:172–203

  MathSciNet  MATH  Google Scholar 

• Babuska I, Suri M (1989) The treatment of the nonhomogeneous Dirichlet boundary conditions by the \(p-\)version of the finite element method. Impact Comput Sci Eng 1:36–63

  MATH  Google Scholar 

• Babuska I, Suri M (1994) The \(p\) and \(h-p\) versions of the finite element method, basic principles and properties. SIAM Rev 36(4):578–632

  MathSciNet  MATH  Google Scholar 

• Balyan L, Dutt P (2012) \(hp\)- spectral element method for elliptic eigenvalue problems. LAP Lambert Academic Publishing

  MATH  Google Scholar 

• Balyan L, Lamba S (2015) Rate of convergence estimates for second order elliptic eigenvalue problems on polygonal domains using spectral element methods. J Numer Anal Ind Appl Math 9–10(1–2):1–10

  MathSciNet  Google Scholar 

• Balyan L, Dutt P, Rathore RKS (2012) Least-squares h-p spectral element method for elliptic eigenvalue problems. Appl Math Comput 218(19):9596–9613

  MathSciNet  MATH  Google Scholar 

• Banerjee U, Suri M (1992) The effect of the numerical quadrature in the \(p-\)version of the finite element method. Math Comput 59(199):1–20

  MathSciNet  MATH  Google Scholar 

• Bangerth W, Rannacher R (2003) Adaptive finite element methods for differential equations. Birkhauser

• Becker EB, Carey GF, Oden JT (1984) Finite elements: computational aspects. Prentice Hall

• Biswas P, Kishore Kumar N, Kar Anil Kumar (2019a) Performance of space-time coupled least-squares spectral element methods for parabolic problems. IOP Conf Ser J Phys Conf Ser 1132:012020

• Biswas P, Dutt P, Ghorai S, Kishore Kumar N (2019b) Space-time coupled least-squares spectral element methods for parabolic problems. Int J Comput Methods Eng Sci Mech 20(5):358–371

• Bochev PB, Gunzburger MD (2009) Least-squares finite element methods, applied mathematical sciences. Springer

• Bramble JH, King JT (1996) A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv Comput Math 6:109–138

  MathSciNet  MATH  Google Scholar 

• Brenner SC, Scott LR (1994) The mathematical theory of finite element methods. Springer Verlag

• Caglar H, Ozer M, Caglar N (2008) The numerical solution of the one-dimensional heat equation by using third degree B-spline functions. Chaos, Solitons Fractals 38:1197–1201

  MATH  Google Scholar 

• Canuto C, Hussaini MY, Quarteroni A, Zang ThA (2006) Spectral methods. In: Fundamentals in single domains. Springer

• Cao Y, Gunzburger MD (1998) Least-squares finite element approximations to solutions of interface problems. SIAM J Numer Anal 35(1):393–405

  MathSciNet  MATH  Google Scholar 

• Carey GF (1989) Parallel supercomputing: methods, algorithms and applications. Wiley

• Carey GF (1997) Computational grids: generation, adaption and solution strategies. Taylor and Francis

• Chen Z, Zou J (1998) Finite element methods and their convergence for elliptic and parabolic interface problems. Numer Math 79:175–202

  MathSciNet  MATH  Google Scholar 

• Ciarlet PG (2002) The finite element method for elliptic problems. SIAM

• Demkowicz L (2007) Computing with \(hp\)-adaptive finite elements, vol. 1, one and two dimensional elliptic and Maxwell problems. CRC Press

• Demkowicz L, Kurtz J, Pardo D (2007) Computing with \(hp\)-adaptive finite elements, vol. 2, frontiers three dimensional elliptic and Maxwell problems with applications. CRC Press

• Di Pietro D, Em A (2012) Mathematical aspects of discontinuous Galerkin methods. Springer

• Dolejsi V, Feistaurer M (2015) Discontinuous Galerkin method. Springer

• Dutt PK, Bedekar S (2001) Spectral methods for hyperbolic initial value problems on parallel computers. J Comput Appl Math 134:165–190

  MathSciNet  MATH  Google Scholar 

• Dutt P, Lamba SS (2009) Domain decomposition methods for hyperbolic problems. Proc Indian Acad Sci 119(2):231–249

  MathSciNet  MATH  Google Scholar 

• Dutt P, Tomar S (2003) Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains. Proc Indian Acad Sci (Math Sci) 113(4):395–429

  MathSciNet  MATH  Google Scholar 

• Dutt P, Tomar SK (2006) A parallel \(hp\)-spectral element method for elliptic problems on polygonal domains-mixed boundary conditions, industrial mathematics. Narosa Publications, pp 121–158

  Google Scholar 

• Dutt P, Tomar S, Kumar BVR (2002) Stability estimates for h-p spectral element methods for elliptic problems. Proc Indian Acad Sci (Math Sci) 112(4):601–639

  MathSciNet  MATH  Google Scholar 

• Dutt PK, Kishore Kumar N, Upadyay CS (2007a) Nonconforming h-p spectral element methods for elliptic problems. Proc Math Sci 117(1):109–145

• Dutt P, Biswas P, Ghorai S (2007b) Spectral element methods for parabolic problems. J Comput Appl Math 203(2):461–486

• Dutt P, Biswas P, Naga Raju G (2008) Preconditioners for spectral element methods for elliptic and parabolic problems. J Comput Appl Math 215:152–166

  MathSciNet  MATH  Google Scholar 

• Dutt P, Akhlaq Husain AS, Murthy Vasudeva, Upadhyay CS (2015a) hp Spectral element methods for three dimensional elliptic problems on non-smooth domains. Appl Math Comput 234:13–35

• Dutt P, Husain A, Vasudeva Murthy AS, Upadhyay CS (2015b) h-p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-I: regularity estimates and stability theorem. Proc Math Sci 125(3):413–447

• Dutt P, Husain A, Vasudeva Murthy AS, Upadhyay CS (2015c) h-p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-II: proof of stability theorem. Proc Math Sci 125(2):239–270

• Dutt P, Husain A, Vasudeva Murthy AS, Upadhyay CS (2016) hp spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-III: error estimates, preconditioners, computational techniques and numerical results. Comput Math Appl 71(9):1745–1771

  MathSciNet  MATH  Google Scholar 

• Epshteyn Y, Phippen S (2015) High-order difference potentials methods for 1 D elliptic type models. Appl Numer Math 93:69–86

  MathSciNet  MATH  Google Scholar 

• Feng X, Karakashian O, Xing Y (2014) Recent developments in discontinuous Galerkin finite element methods for partial differential equations. Springer

• Hansbo A, Hansbo P (2002) An unfitted finite element method based on Nitsche’s method for elliptic interface problems. Comput Methods Appl Mech Eng 191(47–48):5537–5552

  MathSciNet  MATH  Google Scholar 

• Hesthavan JS, Warburton T (2008) Nodal discontinuous Galerkin methods: algorithms, analysis and applications. Springer

• Hon YC, Schaback R, Zhong M (2014) The meshless Kernel-based method of lines for parabolic equations. Comput Math Appl 68:2057–2067

  MathSciNet  MATH  Google Scholar 

• Ji H, Zhang Q, Zhang B (2018) Inf-sup stability of Petrov-Galerkin immersed finite element methods for one-dimensional elliptic interface problems. Numer Methods Part Differ Eqs 34:1917–1932

  MathSciNet  MATH  Google Scholar 

• Jiang B (2010) The least-squares finite element method, theory and applications in CFD and electromagnetics. Springer, Berlin

• Johnson C (1987) Numerical solutions of partial differential equations by finite element method. Cambridge University Press

• Jones D, Zhang Xu (2019) A high order immersed finite element method for parabolic interface problems. ITM Web Conf 29:01007

• Kandilarov JD (2004) Immersed boundary level set approach for numerical solution for elliptic interface problems. In: LSSC 2003. Lect Notes Comput Sci 2907:96

• Kaneko H, Bey Kim S, Hou GJW (2006) Discontinuous Galerkin finite element method for parabolic problems. Appl Math Comput 182(1):388–402

  MathSciNet  MATH  Google Scholar 

• Karniadakis G, Spencer S (2005) Spectral/hp element methods for computational fluid dynamics. Oxford

• Khan A, Hussain A (2017a) Exponentially accurate spectral element method for fourth order elliptic problems. J Sci Comput 71(1):303–328

• Khan A, Hussain A (2017b) Least-squares spectral element preconditioners for fourth order elliptic problems. Comput Math Appl 74(3):482–503

• Khan A, Upadhyay CS (2016) Exponentially accurate nonconforming least-squares spectral element method for elliptic problems on unbounded domains. Comput Methods Appl Mech Eng 305:607–633

  MathSciNet  MATH  Google Scholar 

• Khan A, Dutt P, Upadhyay CS (2015) Nonconforming least-squares spectral element method for European options. Comput Math Appl 70(1):47–65

  MathSciNet  MATH  Google Scholar 

• Khan A, Hussain A, Mohapatra S, Upadhyay CS (2017) Spectral element method for three dimensional elliptic problems with smooth interfaces. Comput Methods Appl Mech Eng 315:522–549

  MathSciNet  MATH  Google Scholar 

• Khan A, Upadhyay CS, Gerritsma M (2018) Spectral element method for parabolic interface problems. Comput Methods Appl Mech Eng 337:66–94

  MathSciNet  MATH  Google Scholar 

• Kishore Kumar N (2014) Nonconforming spectral element method for elasticity interface problems. J Appl Math Inf 32(5–6):761–781

  MathSciNet  MATH  Google Scholar 

• Kishore Kumar N, Biswas Pankaj (2021) Fully discrete least-squares spectral element method for parabolic interface problems. Math Comput Simul 181:364–379

  MathSciNet  MATH  Google Scholar 

• Kishore Kumar N, Mohapatra Shubhashree (2022) Performance of nonconforming spectral element method for Stokes problems. Comput Appl Math 41:156

• Kishore Kumar N, Naga Raju G (2010) Least-squares hp/spectral element method for elliptic problems. Appl Numer Math 60:38–54

  MathSciNet  MATH  Google Scholar 

• Kishore Kumar N, Naga Raju G (2012) Nonconforming least-squares method for elliptic partial differential equations with smooth interfaces. J Sci Comput 53(2):295–319

  MathSciNet  MATH  Google Scholar 

• Kishore Kumar N, Dutt P, Upadyay CS (2009) Nonconforming spectral/\(hp\) element methods for elliptic systems. J Numer Math 17(2):119–142

  MathSciNet  MATH  Google Scholar 

• Kishore Kumar N, Biswas Pankaj, Seshadri Reddy B (2020) A study of spectral element method for elliptic interface problems with nonsmooth solutions in \({\mathbb{R} }^{2},\). J Appl Math Inf 38(3–4):311–334

  MATH  Google Scholar 

• Knabber P, Angermann L (2003) Numerical methods for elliptic and parabolic partial differential equations. Springer

• Kopriva DA (2009) Implementing spectral methods for partial differential equations. In: Algorithms for scientists and engineers. Springer, 293–354

• Kumar M, Joshi P (2012) Some numerical techniques for solving elliptic interface problems. Numer Methods Partial Differ Equ 28(1):94–114

  MathSciNet  MATH  Google Scholar 

• Lacour C, Belgacem FB (2012) The mortar finite element method: basics, theory and implementation. CRC Press

• Lamba S, Dutt P (2011) Pulsating spectral element method for hyperbolic problems. VDM Verlag

• Langtangen HP, Linge S (2017) Finite difference computing with PDE’s, a modern software approach. Springer

• Larson MG, Bengzon F (2013) The finite element method, theory, implementation and applications. Springer

• Lee U (2009) Spectral element method in structural dynamics. J. Wiley

• Lehrenfeldand C, Reusken A (2018) Analysis of a high-order unfitted finite element method for elliptic interface problems. IMA J Numer Anal 38(3):1351–1387

  MathSciNet  MATH  Google Scholar 

• LeVeque RJ (2007) Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. SIAM

• Li Z (1998) The immersed interface method using finite element formulation. Appl Numer Math 27:255–267

  MathSciNet  Google Scholar 

• Li Z, Ito K (2006) The immersed interface method, numerical solutions of PDEs involving interfaces and irregular domains. SIAM

• Li Z, Ji H, Chen X (2017) Accurate solutions and gradient computation for elliptic interface problems with variable coefficients. SIAM J Numer Anal 55(2):570–597

  MathSciNet  MATH  Google Scholar 

• Lin T, Lin Y, Sun W (2007) Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete Contin Dyn Syst Ser B 7(4):807–823

  MathSciNet  MATH  Google Scholar 

• Loubenets A, Ali T, Hanke M (2009) Highly accurate finite element method for one-dimensional elliptic interface problems. Appl Numer Math 59(1):119–134

  MathSciNet  MATH  Google Scholar 

• Maday Y, Patera AT (1989) Spectral element methods for the incompressible Navier-Stokes equations. In: Noor AK (ed) State of the art survey on computational mechanics. ASME, New York

• Mathew T (2008) Domain decomposition methods for the numerical solution of partial differential equations. In: Lecture notes in computational science and engineering, Springer

• Mohapatra S, Ganesan S (2016) A non-conforming least squares spectral element formulation for Oseen equations with applications to Navier-Stokes equations. Numer Funct Anal Optim 37:1295–1311

  MathSciNet  MATH  Google Scholar 

• Mohapatra S, Hussain A (2016) Least-squares spectral element method for three dimensional Stokes equations. Appl Numer Math 102:31–54

  MathSciNet  MATH  Google Scholar 

• Mohapatra S, Dutt PK, Rathish Kumar BV, Gerritsma Marc I (2020) Non-conforming least squares spectral element method for Stokes equations on non-smooth domains. J Comput Appl Math 372:112696

  MathSciNet  MATH  Google Scholar 

• Naga Raju G, Dutt P, Kishore Kumar N, Upadhyay CS (2014) Spectral element method for elliptic equations with periodic boundary conditions. Appl Math Comput 246:426–439

  MathSciNet  MATH  Google Scholar 

• Ozisik MN, Oralnde HRB, Colaco MJ, Colta RM (1994) Finite difference methods in heat transfer, 2nd edn. CRC Press

  Google Scholar 

• Pan K, Tan Y, Hu H (2010) An interpolation matched interface and boundary method for elliptic interface problems. J Comput Appl Math 234:73–94

  MathSciNet  MATH  Google Scholar 

• Patera AT (1984) A spectral element method for fluid dynamics—Laminar flow in a channel expansion. J Comput Phys 54:468–488

  MATH  Google Scholar 

• Pathria D, Karniadakis GE (1995) Spectral element methods for elliptic problems in nonsmooth domains. J Comput Phys 122:83–95

  MathSciNet  MATH  Google Scholar 

• Pozrikidis C (2014) Introduction to finite and spectral element methods using MATLAB. CRC Press

• Quarteroni A (2014) Numerical models for differential problems. Springer

• Reddy JN (2005) An introduction to the finite element method. McGraw Hill

• Riviere B (2008) Discontinuous Galerkin methods for solving elliptic and parabolic equations. Frontiers in Applied Mathematics. SIAM

• Schwab CH (1998) \(p\) and \(h-p\) finite element methods. Clarendor Press, Oxford

• Shin BC, Jung JH (2011) Spectral collocation and radial basis function methods for one dimensional interface problems. Appl Numer Math 61:911–928

  MathSciNet  MATH  Google Scholar 

• Smith IM, Griffiths DV, Margetts L (2013) Programming the finite element method. Wiley

• Solin P, Segeth K, Dolezel I (2003) High-order finite element methods. CRC Press

• Strang G, Fix G (2008) An analysis of the finite element method. Wellesley-Cambridge Press

• Szabo B, Babuska I (1991) Finite element analysis. Wiley

• Thomas JW (1995) Numerical partial differential equations, finite difference methods. Springer

• Thomee V (2006) Galerkin finite element methods for parabolic problems, 2nd edn. Springer

• Tomar SK (2006) \(h-p\) Spectral element method for elliptic problems on non-smooth domains using parallel computers. Computing 78:117–143

  MathSciNet  MATH  Google Scholar 

• Tomar SK, Dutt P, Rathish Kumar BV (2002a) An efficient and exponentially accurate parallel hp spectral element method for elliptic problems on polygonal domains—the Dirichlet case. Lect Notes Comput Sci 2552:534–544

• Tomar SK, Dutt P, Rathish Kumar BV (2002b) Parallel \(hp-\)spectral element method for elliptic problems on polygonal domains. In: Euro Conference on numerical mathematics and computational mechanics, NMCM

• Toselli A, Widlund O (2005) Domain decomposition methods, algorithms and theory. Springer

• Trangenstein JA (2013) Numerical solution of elliptic and parabolic differential equations. Cambridge University Press

• Trefethen LN (2000) Spectral methods in MATLAB. SIAM

• Zhang Q, Weng Z, Ji H, Jhang B (2018) Error estimates for an augmented method for one-dimensional elliptic interface problems. Adv Differ Equ 2018:307

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