A POD-based reduced-order TSCFE extrapolation iterative format for two-dimensional heat equations
- Zhendong Luo^{1}Email author
https://doi.org/10.1186/s13661-015-0320-x
© Luo; licensee Springer. 2015
Received: 19 October 2014
Accepted: 23 March 2015
Published: 4 April 2015
Abstract
In this article, a proper orthogonal decomposition (POD) technique is employed to establish a POD-based reduced-order time-space continuous finite element (TSCFE) extrapolation iterative format for two-dimensional (2D) heat equations, which includes very few degrees of freedom but holds sufficiently high accuracy. The error estimates of the POD-based reduced-order TSCFE solutions and the algorithm implementation of the POD-based reduced-order TSCFE extrapolation iterative format are provided. A numerical example is used to illustrate that the results of the numerical computation are consistent with the theoretical conclusions. Moreover, it is shown that the POD-based reduced-order TSCFE extrapolation iterative format is feasible and efficient for solving 2D heat equations.
Keywords
MSC
1 Introduction
The time-space finite element (FE) methods for time-dependent partial differential equations (TDPDEs) play an important role in many practical applications and form an important research topic (see [1–7]). They have a higher accuracy than their usual FE methods with time-forward difference and even they have a higher accuracy than their Crank-Nicolson FE methods with time-averaged data (see, e.g., [8–10]). However, even if the time-space continuous finite element (TSCFE) methods for two-dimensional (2D) heat equations include a lot of degrees of freedom too, they would cause many difficulties for real-life applications. Therefore, an important problem is how as few it is possible to use to lessen the degrees of freedom so as to alleviate the computational load and save time-consuming calculations and resource demands in the computational process in a way that guarantees the sufficiently accurate numerical solutions.
The proper orthogonal decomposition (POD) method (see [11]) is an efficient means to lessen the degrees of freedom of numerical models for TDPDEs and alleviate the accumulation of truncation errors in the computational process so as to reduce the computational load and save memory requirements. It has been widely and successfully applied to numerous fields, including signal analysis and pattern recognition (see [12]), statistics, geophysical fluid dynamics or meteorology (see [13]). It essentially provides an orthogonal basis for representing the given data in a certain least squares optimal sense, namely, it provides a way to find optimal lower dimensional approximations of the given data. Especially, it has played an important role in reduced-order of numerical methods for TDPDEs (see, e.g., [14–31]). Moreover, the long-term stability of POD reduced-order models is discussed (see, e.g., [32–35]).
However, almost all existing POD-based reduced-order numerical methods (see, e.g., [14–35]) employ numerical solutions obtained from classical numerical methods on the total time span \([0,T]\) to form POD bases and establish reduced-order models, and then recompute the solutions on the same time span \([0,T]\), which actually entails repeating computations on the same time span \([0,T]\). Especially, to the best of our knowledge, there is not any report that the POD-based reduced-order TSCFE extrapolation iterative format for 2D heat equations is established or that an algorithm of the reduced-order TSCFE extrapolation iterative format is implemented. Therefore, in this article, we establish the POD-based reduced-order TSCFE extrapolation iterative format for 2D heat equations and provide the error estimates of the POD-based reduced-order TSCFE solutions and the algorithm implementation of the POD-based reduced-order TSCFE extrapolation iterative format. We also provide a numerical example to illustrate that the POD-based reduced-order TSCFE extrapolation iterative format is feasible and efficient for seeking numerical solutions for 2D heat equations. Especially, we here thoroughly improve the existing methods, namely, we do only employ the first few given classical TSCFE solutions on a very short time span \([0, T_{0}]\) (\(T_{0}\ll T\)) as snapshots to form POD basis and establish the POD-based reduced-order TSCFE extrapolation iterative format for seeking the numerical solutions on the total time span \([0, T]\). Thus, we can sufficiently absorb the advantage of the POD method, namely, sufficiently utilize the given data (on very short time span \([0, T_{0}]\) and \(T_{0}\ll T\)) to forecast future physical phenomena (on the time span \([T_{0},T]\)). Therefore, the POD-based reduced-order TSCFE extrapolation iterative format for 2D heat equations is completely different from the existing POD-based reduced-order methods (see, e.g., [14–35] etc.) and we have an improvement and development of the existing methods as mentioned above or others.
The article is organized as follows. Section 2 recalls the classical TSCFE method for 2D heat equations and generates snapshots by means of the first fewer TSCFE solutions obtained from the classical TSCFE formulation. In Section 3, we form orthonormal POD bases of a set of the snapshots by means of the POD method and establish the POD-based reduced-order TSCFE extrapolation iterative format including very few degrees of freedom but holding sufficiently high accuracy for the 2D heat equations. In Section 4, the error estimates of the POD-based reduced-order TSCFE solutions and the algorithm implementation for the POD-based reduced-order TSCFE extrapolation iterative format are provided. In Section 5, a numerical example is presented to illustrate that the results of the numerical computation are consistent with the theoretical conclusions and validate that the POD-based reduced-order TSCFE extrapolation iterative format is feasible and efficient for finding numerical solutions to the 2D heat equations and can greatly lessen its degrees of freedom and alleviate the computational load as well as save time for calculations and resource demands in the computational process. Section 6 provides the main conclusions and discussions.
2 Recall the classical TSCFE method for 2D heat equations and generate snapshots
Let \(\Omega\subset\mathbf{R}^{2}\) be a bounded and connected polygonal domain. Consider the following 2D heat equations (see [11, 12]).
Problem I
Problem II
Problem III
The following results of the existence, the uniqueness, the stability, and the convergence of the solution for the system of equations (2.14) and (2.15), namely, Problem III, are obtained by means of the same methods as the proofs of Theorems 3.1, 3.2, 3.3, and 4.4 in [1] or the analogous approaches of proving Theorems 3.1, 4.1, 4.2, and 5.1 in [2].
Theorem 2.1
Remark 2.2
If the source term \(f(x,y,t)\), the initial value function \(\varphi_{0}(x, y)\), the time step k, and the spatial mesh size h all are given, then we can obtain solutions \(u^{hk}(x,y,t)\) by solving Problem III or the system of equations (2.14) and (2.15). But we obtain the first M solutions \(u^{n,0}(x,y)=u(x,y,t_{n})\) (\(1\le n\le M\)) (in general, \(M \ll N\), for example, \(M =20\), \(N=1\text{,}000\)) by solving (2.15) at \(n=1, 2, \ldots, M\), which are referred to as snapshots. When one computes actual problems, one may obtain the ensemble of snapshots from physical system trajectories by drawing samples from experiments.
3 Form POD bases and establish POD-based reduced-order TSCFE extrapolation iterative format
In this section, we refer to the idea in [15, 24, 26, 27] to form a POD basis (for more details see [15, 24, 26, 27]) and establish the POD-based reduced-order TSCFE extrapolation iterative format for 2D heat equations.
Form the matrix \(\mathbf{A}=(A_{ij})_{M\times M}\in R^{ M\times M}\), where \({A}_{ij}=(\nabla u^{i,0}, \nabla u^{j,0})/M\) and \(u^{n,0}(x,y)\) (\(n=1, 2, \ldots, M\)) are the snapshots extracted in Section 2. Since the matrix A is positive semi-definite with rank \(\kappa =\dim(\operatorname{span}\{u^{n,0}(x,y); n=1,2, \ldots, M\})\), its positive eigenvalues and the corresponding orthonormal eigenvectors are used to form POD bases as follows [15, 24, 26, 27].
Lemma 3.1
Lemma 3.2
Thus, based on \(S_{dm}(\Omega)\), the POD-based reduced-order TSCFE extrapolation iterative format for the 2D heat equations is established as follows.
Problem IV
Remark 3.3
If \(\Im_{h}\) is a uniformly regular triangulation, even though \(S_{hm}(\Omega)\) is the spaces of piecewise linear polynomials, i.e., \(m=1\), the number of total degrees of freedom for Problem III on each time level is \(N_{h}\times l\) (where \(N_{h}\) is the number of vertices of triangles in \(\Im_{h}\)). If \(m=2\), the number of total degrees of freedom for Problem III on each time level is \(4N_{h}\times l\), while the number of total degrees of freedom for Problem IV on each time level is \(d\times l\) (\(d\ll\kappa\le M \ll N\ll N_{h}\)). For scientific engineering problems, the number \(N_{h}\) of vertices of triangles in \(\Im_{h}\) is more than tens of thousands, even more than a hundred million, while d is only the number of the first few main eigenvalues so that it is very small (for example, in Section 5, \(d=6\), while \(N_{h}>2\times10^{2}\times2\times10^{2}=4\times 10^{4}\)). Therefore, Problem IV is the POD-based reduced-order TSCFE extrapolation iterative format with very few degrees of freedom for the 2D heat equations. Especially, it has no repeating computations and uses the given solutions on the first fewer M time steps for Problem III to extrapolate other \((n-M)\) solutions, which is completely different from existing reduced-order approaches (see, e.g., [14–35] etc.).
4 Error analysis and algorithm implementation of POD-based reduced-order TSCFE extrapolation iterative format
4.1 Error estimates of solutions for Problem IV
In the following, we employ classical TSCFE method to derived the error estimates of the POD-based reduced-order TSCFE solutions for Problem IV. We have the following main results for Problem IV.
Theorem 4.1
Proof
The solutions of Problem IV on \(J_{n}\) (\(n=1,2, \ldots, M\)) are obviously unique since they are obtained by projecting the solutions of (2.14) and (2.15), namely Problem III on \(J_{n}\) (\(n=1,2, \ldots, M\)) into POD basis. Moreover, (4.1) and (4.3) are immediately obtained from (3.6)-(3.7) and Theorem 2.1.
With Theorem 2.1 and by means of the analogous proof of Theorem 4.1, we easily obtain the following corollary.
Corollary 4.2
Remark 4.3
Due to POD-based reduced-order and extrapolation for the classical TSCFE formulation Problem III, the errors of the solutions for the POD-based reduced-order TSCFE extrapolation iterative format Problem IV include more term \((k^{{-1}/{2}}h^{2}\sum_{i=d+1}^{\kappa}\lambda_{j} )^{1/2}\) than those for Problem III, but the degrees of freedom for Problem IV are far less than those of Problem III so that Problem IV can greatly lessen the truncation error accumulation in the computational process, alleviate the calculating load, save time-consuming calculations, and improve actual computational accuracy (see the example in Section 5). However, the factor \((k^{{-1}/{2}}h^{2}\sum_{i=d+1}^{\kappa}\lambda_{j})^{{1}/{2}}\) in Theorem 4.1 and Corollary 4.2 can act as a suggestion to choose the number of POD bases, namely, it is only necessary to choose d such that \((k^{{-1}/{2}}h^{2}\sum_{i=d+1}^{\kappa}\lambda_{j})^{{1}/{2}}=O(k^{l},h^{m})\).
4.2 Algorithm implementation of POD-based reduced-order TSCFE extrapolation iterative format
Finding the solutions of the POD-based reduced-order TSCFE extrapolation iterative format for 2D heat equations consists of the following six steps.
Step 1. For given initial value function \(\varphi_{0}(x,y)\), source term \(f(x,y,t)\), and the time step increment k and the spatial grid measurement h, solving (2.14) at the first M steps obtains the snapshots \(u^{n,0}\) (\(n=1,2,\ldots, M\)).
Step 2. Form the correlation matrix \(\mathbf{A}=(A_{ij})_{M\times M}\in R^{ M\times M}\), where \({A}_{ij}=(\nabla u^{i,0}, \nabla u^{j,0})/M\) and \((\cdot,\cdot)\) is \(L^{2}\)-inner product.
Step 3. Let \({\mathbf{v}}=(a_{1}, a_{2}, \ldots, a_{M})^{T}\). Solving the eigenvalue problem \({\mathbf{A}\mathbf{v}}=\lambda{\mathbf{v}}\) yields positive eigenvalues \(\lambda_{1}\ge\lambda_{2}\ge\cdots\ge\lambda _{\kappa}>0\) (\(\kappa=\dim(\operatorname{span}\{u^{n,0}(x,y); n=1,2, \ldots, M\})\)) and the corresponding eigenvectors \(\mathbf{v}^{j}=(a_{1}^{j},a_{2}^{j},\ldots, a_{M}^{j})^{T}\) (\(j=1,2,\ldots,\kappa\)).
Step 4. For the error \(\delta=O(k^{l}, h^{m})\) needed, determine the number d of POD basis such that \(k^{-1/{2}}h^{2}\sum_{j=d+1}^{\kappa}\lambda_{j}\le\delta^{2}\).
Step 5. Generate POD basis \({\psi}_{i} =\sum_{j=1}^{M} a_{j}^{i}u^{j,0}(x,y)/{\sqrt{M\lambda_{i}}}\) (\(j=1,2,\ldots,d\)). Let \(S_{dm}(\Omega)=\operatorname{span}\{\psi _{1}, \psi_{2}, \ldots, \psi_{d} \}\). Solving Problem IV with d degrees of freedom obtains \(u^{dk}(x,y,t)\).
Step 6. If \(\|\nabla(u_{d}^{n-1,0}- u_{d}^{n,0})\|_{0}\ge\| \nabla(u_{d}^{n,0}- u_{d}^{n+1,0})\|_{0}\) (\(n=M, M+1,\ldots, N-1\)), then \(u^{dk}\) are the solutions for Problem IV satisfying accuracy needed. Else, namely, if \(\|\nabla(u_{d}^{n-1,0}- u_{d}^{n,0})\|_{0}<\|\nabla (u_{d}^{n,0}- u_{d}^{n+1,0})\|_{0}\) (\(n=M, M+1,\ldots, N-1\)), let \(u^{i,0}=u^{n-M-i}\) (\(i=1, 2, \ldots, M\)) and return to Step 2.
5 A numerical example
In this section, we present a numerical example showing the advantage of the POD-based reduced-order TSCFE extrapolation iterative format for 2D heat equations.
Further, by comparing the classical TSCFE formulation with the POD-based reduced-order TSCFE extrapolation iterative format with six POD bases implementing the numerical simulations at \(t=10\), it is found that the classical TSCFE formulation includes more than \(2\times16\times 10^{4}\) unknown quantities (since the subspaces \(S_{hm}(\Omega)\) and \(S_{kl}(0,T)\) are taken as piecewise second polynomial spaces) on each time level and the required computing time is 240 minutes, while the POD-based reduced-order TSCFE extrapolation iterative format with six POD bases does only include six unknown quantities on each time level and the corresponding computing time is less than 60 seconds, namely, the computing time of the classical TSCFE formulation is 240 times higher than that of the POD-based reduced-order TSCFE extrapolation iterative format with six POD bases. Thus, the POD-based reduced-order TSCFE extrapolation iterative format can greatly save the calculating time and alleviate the computing load so that it could greatly lessen the truncation error accumulation in the computational process. It is also shown that finding the numerical solutions for 2D heat equations by means of the POD-based reduced-order TSCFE extrapolation iterative format is computationally very effective and feasible.
6 Conclusions and discussions
In this article, the POD-based reduced-order TSCFE extrapolation iterative format for 2D heat equations has been established, the error estimates of the POD-based reduced-order TSCFE solutions and the algorithm implementation of the POD-based reduced-order TSCFE extrapolation iterative format have been provided. A numerical example has been used to validate that the results of the numerical computation are consistent with the theoretical conclusions. Further, it has shown that the POD-based reduced-order TSCFE extrapolation iterative format can greatly save the calculating time and alleviate the computing load so that it could greatly lessen the truncation error accumulation in the computational process. Moreover, it is shown that the POD-based reduced-order TSCFE extrapolation iterative format is computationally very effective and feasible for finding the numerical solutions to 2D heat equations.
Though some POD-based reduced-order models for 2D parabolic equations have been established (see [15, 19, 23, 27, 31]), the POD-based reduced-order TSCFE extrapolation iterative format here is completely different from those mentioned above because the TSCFE method is different from the usual Galerkin method, the FE method, the finite volume method, and the finite difference scheme. Especially, the POD-based reduced-order TSCFE extrapolation iterative format only employs the first few given classical TSCFE solutions on the very short time span \([0, T_{0}]\) (\(T_{0}\ll T\)) as snapshots to form the POD basis and seek the numerical solutions on the total time span \([0, T]\). It is an improvement and innovation of the existing methods (see, e.g., [14–35] etc.).
Future research work in this area will aim to extend the POD-based reduced-order TSCFE extrapolation iterative format, applying it to more TDPDEs such as the nonlinear Schrödinger equation, integral differential equations, convection-diffusion equations, nonlinear sine-Gordon equation, Sobolev equations, solute transport equations, and viscoelastic equations.
Declarations
Acknowledgements
This research was supported by National Science Foundation of China grant 11271127 and Science Research Project of Guizhou Province Education Department grant QJHKYZ[2013]207.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Aziz, AK, Monk, P: Continuous finite elements in space and time for the heat equation. Math. Comput. 52, 255-274 (1989) View ArticleMATHMathSciNetGoogle Scholar
- Karakashia, O, Makkridakis, C: A time-space finite element method for nonlinear Schrödinger equation: the continuous Galerkin method. Math. Comput. 97(222), 479-499 (1998) View ArticleGoogle Scholar
- Li, H, Liu, RX: The space-time finite element method for parabolic problems. Appl. Math. Mech. 22(6), 687-700 (2001) View ArticleMATHMathSciNetGoogle Scholar
- Bales, L, Lasiecka, I: Continuous finite elements in space and time for the nonhomogeneous wave equation. Comput. Math. Appl. 27(3), 91-102 (1994) View ArticleMATHMathSciNetGoogle Scholar
- French, DA, Peterson, TE: A continuous space-time finite element method for the wave equation. Math. Comput. 65(214), 491-506 (1996) View ArticleMATHMathSciNetGoogle Scholar
- Liu, Y, Li, H, He, S: Mixed time discontinuous space-time finite element method for convection diffusion equations. Appl. Math. Mech. 29(12), 1579-1586 (2008) View ArticleMATHMathSciNetGoogle Scholar
- French, DA: A space-time finite element method for the wave equation. Comput. Methods Appl. Mech. Eng. 107, 145-157 (1993) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD: The Foundations and Applications of Mixed Finite Element Methods. Chinese Science Press, Beijing (2006) (in Chinese) Google Scholar
- Thomée, V: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, Berlin (2003) MATHGoogle Scholar
- Thomée, V: Negative norm estimates and superconvergence in Galerkin methods for parabolic problems. Math. Comput. 34, 93-113 (1980) View ArticleMATHMathSciNetGoogle Scholar
- Holmes, P, Lumley, JL, Berkooz, G: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996) View ArticleMATHGoogle Scholar
- Fukunaga, K: Introduction to Statistical Recognition. Academic Press, New York (1990) MATHGoogle Scholar
- Jolliffe, IT: Principal Component Analysis. Springer, Berlin (2002) MATHGoogle Scholar
- Ly, HV, Tran, HT: Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Q. Appl. Math. 60, 631-656 (2002) MATHMathSciNetGoogle Scholar
- Kunisch, K, Volkwein, S: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 117-148 (2001) View ArticleMATHMathSciNetGoogle Scholar
- Kunisch, K, Volkwein, S: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492-515 (2002) View ArticleMATHMathSciNetGoogle Scholar
- Kunisch, K, Volkwein, S: Control of Burgers’ equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102, 345-371 (1999) View ArticleMATHMathSciNetGoogle Scholar
- Ahlman, D, Södelundon, F, Jackson, J, Kurdila, A, Shyy, W: Proper orthogonal decomposition for time-dependent lid-driven cavity flows. Numer. Heat Transf., Part B, Fundam. 42, 285-306 (2002) View ArticleGoogle Scholar
- Luo, ZD, Chen, J, Xie, ZH, An, J, Sun, P: A reduced second-order time accurate finite element formulation based on POD for parabolic equations. Sci. Sin., Math. 41(5), 447-460 (2011) (in Chinese) View ArticleGoogle Scholar
- Luo, ZD, Li, H, Zhou, YJ, Huang, XM: A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem. J. Math. Anal. Appl. 385, 371-383 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD, Li, H, Zhou, YJ, Xie, ZH: A reduced finite element formulation and error estimates based on POD method for two-dimensional solute transport problems. J. Math. Anal. Appl. 385, 371-383 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD, Du, J, Xie, ZH, Guo, Y: A reduced stabilized mixed finite element formulation based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations. Int. J. Numer. Methods Eng. 88(1), 31-46 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD, Xie, ZH, Shang, YQ, Chen, J: A reduced finite volume element formulation and numerical simulations based on POD for parabolic equations. J. Comput. Appl. Math. 235(8), 2098-2111 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD, Xie, ZH, Chen, J: A reduced MFE formulation based on POD for the non-stationary conduction-convection problems. Acta Math. Sci. 31(5), 1765-1785 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Li, HR, Luo, ZD, Chen, J: Numerical simulation based on proper orthogonal decomposition for two-dimensional solute transport problems. Appl. Math. Model. 35(5), 2489-2498 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD, Zhou, YJ, Yang, XZ: A reduced finite element formulation based on proper orthogonal decomposition for Burgers equation. Appl. Numer. Math. 59(8), 1933-1946 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD, Chen, J, Sun, P, Yang, XZ: Finite element formulation based on proper orthogonal decomposition for parabolic equations. Sci. Sin., Math. 52(3), 587-596 (2009) MathSciNetGoogle Scholar
- Luo, ZD, Chen, J, Navon, IM, Yang, XZ: Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations. SIAM J. Numer. Anal. 47(1), 1-19 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD, Chen, J, Navon, IM, Zhu, J: An optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems. Int. J. Numer. Methods Fluids 60(4), 409-436 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD, Zhu, J, Wang, RW, Navon, IM: Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model. Comput. Methods Appl. Mech. Eng. 196(41-44), 4184-4195 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Luo, ZD, Li, H, Shang, YQ, Fang, Z: A LSMFE formulation based on proper orthogonal decomposition for parabolic equations. Finite Elem. Anal. Des. 60, 1-12 (2012) View ArticleMathSciNetGoogle Scholar
- Bergmann, M, Bruneau, CH, Iollo, A: Enablers for robust POD models. J. Comput. Phys. 228(2), 516-538 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Deane, AE, Kevrekidis, IG, Karniadakis, GE, Orsag, SA: Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinder. Phys. Fluids A 3(10), 2337-2354 (1991) View ArticleMATHGoogle Scholar
- Ma, X, Karniadakis, GE: A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458, 181-190 (2002) View ArticleMATHMathSciNetGoogle Scholar
- Wang, Z, Akhtar, I, Borggaard, J, Iliescu, T: Two-level discretizations of non-linear closure models for proper orthogonal decomposition. J. Comput. Phys. 230, 126-146 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Adams, RA: Sobolev Spaces. Academic Press, New York (1975) MATHGoogle Scholar
- Brezzi, F, Fortin, M: Mixed and Hybrid Finite Element Methods. Springer, New York (1991) View ArticleMATHGoogle Scholar
- Ciarlet, PG: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) MATHGoogle Scholar
- Rudin, W: Functional Analysis, 2nd edn. McGraw-Hill, New York (1973) MATHGoogle Scholar