[go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Block Preconditioning Methods for Asymptotic Preserving Scheme Arising in Anisotropic Elliptic Problems

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Efficient and robust iterative solvers for strongly anisotropic elliptic equations are very challenging. Indeed, the discretization of this class of problems gives rise to a linear system with a condition number increasing with anisotropic strength. This weakness is addressed clearly by adopting the asymptotic-preserving (AP) discretizations. In this paper a block preconditioning method is introduced to solve the linear algebraic systems of a class of micro–macro asymptotic-preserving (MMAP) scheme. The MMAP method was developed by Degond et al. in 2012 where its corresponding discrete matrix has a \(2\times 2\) block structure. Motivated by approximate Schur complements, a series of block preconditioners are constructed. We first analyze a natural approximate Schur complement that is the coefficient matrix of the original Non-AP discretization. However it tends to be singular for very small anisotropic parameters. We then improve it by using more suitable approximation for boundary rows of the exact Schur complement. With these block preconditioners, a preconditioned GMRES iterative method is developed to solve the discrete equations. Several numerical tests show that block preconditioning methods can be a practically useful strategy with respect to grid refinement and anisotropic strengths.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data Availability

All data generated or analysed during this study are included in this published article.

References

  1. Ahrbi, B.R., Mavriplis, D.J.: A scalable solution strategy for high-order stabilized finite-element solvers using an implicit line preconditioner. Comput. Methods Appl. Mech. Eng. 341, 956–984 (2018)

    Article  MathSciNet  Google Scholar 

  2. Ahrbi, B.R., Mavriplis, D.J.: An implicit block ILU smoother for preconditioning of Newton–Krylov solvers with application in high-order stabilized finite-element methods. Comput. Methods Appl. Mech. Eng. 358, 112637 (2020)

    Article  MathSciNet  Google Scholar 

  3. Besse, C., Degond, P., Deluzet, F., Claudel, J., Gallice, G., Tessieras, C.: A model hierarchy for ionospheric plasma modeling. Math. Models Methods Appl. Sci. 14(03), 393–415 (2004)

    Article  MathSciNet  Google Scholar 

  4. Besse, C., Deluzet, F., Negulescu, C., Yang, C.: Efficient numerical methods for strongly anisotropic elliptic equations. J. Sci. Comput. 55(1), 231–254 (2013)

    Article  MathSciNet  Google Scholar 

  5. Brown, P.N., Walker, H.F.: GMRES on (nearly) singular systems. SIAM J. Matrix Anal. Appl. 18(1), 37–51 (1997)

    Article  MathSciNet  Google Scholar 

  6. Chacón, L., Del-Castillo-Negrete, D., Hauck, C.D.: An asymptotic-preserving semi-lagrangian algorithm for the time-dependent anisotropic heat transport equation. J. Comput. Phys. 272, 719–746 (2014)

    Article  MathSciNet  Google Scholar 

  7. Degond, P., Deluzet, F., Negulescu, C.: An asymptotic preserving scheme for strongly anisotropic elliptic problems. Multiscale Model. Simul. 8(2), 645–666 (2009)

    Article  MathSciNet  Google Scholar 

  8. Degond, P., Lozinski, A., Narski, J., Negulescu, C.: An asymptotic-preserving method for highly anisotropic elliptic equations based on a Micro–Macro decomposition. J. Comput. Phys. 231(7), 2724–2740 (2012)

    Article  MathSciNet  Google Scholar 

  9. Deluzet, F., Narski, J.: A two field iterated asymptotic-preserving method for highly anisotropic elliptic equations. Multiscale Model. Simul., pp. 434–459 (2019)

  10. Elman, H., Silvester, D.D., Wathen, A.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, New York (2014)

    Book  Google Scholar 

  11. Elman, H., Howle, V.E., Shadid, J., Shuttleworth, R., Tuminaro, R.: A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations. J. Comput. Phys. 227, 1790–1808 (2008)

    Article  MathSciNet  Google Scholar 

  12. Evans, L.C.: Partial Differential Equations. American Mathematical Society (2010)

  13. Fichtner, W., Rose, D.J.: On the numerical solution of nonlinear elliptic pdes arising from semiconductor device modeling. In: Schultz, M.H., (ed) Elliptic Problem Solvers, pp. 277–284. Academic Press (1981)

  14. Fonseca, C.: On the eigenvalues of some tridiagonal matrices. J. Comput. Appl. Math. 200(1), 283–286 (2007)

    Article  MathSciNet  Google Scholar 

  15. Giorgiani, G., Bufferand, H., Schwander, F., Serre, E., Tamain, P.: A high-order non field-aligned approach for the discretization of strongly anistropic diffusion operators in magnetic fusion. Comput. Phys. Commun. 254, 107375 (2020)

    Article  MathSciNet  Google Scholar 

  16. Greenbaum, A., Pták, V., Strakos̆, Z.: Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl., 17(3), 465–469 (1996)

  17. Griffies, S.M., Böning, C., Bryan, F.O., Chassignet, E.P., Gerdes, R., Hasumi, H., Hirst, A., Treguier, A.-M., Webb, D.: Developments in ocean climate modelling. Ocean Model. 2(3), 123–192 (2000)

    Article  Google Scholar 

  18. Hou, T.Y., Wu, X.-H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)

    Article  MathSciNet  Google Scholar 

  19. Krasheninnikova, N., Chacón, L.: Fourth order discretization of anisotropic heat conduction operator. In: APS Division of Plasma Physics Meeting Abstracts, vol. 50 of APS Meeting Abstracts, page BP6.040 (2008)

  20. Li, L., Yang, C.: APFOS-Net: asymptotic preserving scheme for anisotropic elliptic equations with deep neural network. J. Comput. Phys. 453, 110958 (2022)

    Article  MathSciNet  Google Scholar 

  21. Li, Q., Jiang, L.: A multiscale virtual element method for elliptic problems in heterogeneous porous media. J. Comput. Phys. 388, 394–415 (2019)

    Article  MathSciNet  Google Scholar 

  22. Meier, E.T.: Modeling plasmas with strong anisotropy, neutral fluid effects, and open boundaries. PhD thesis, University of Washington (2011)

  23. Meiss, J.D.: Plasma Confinement. Dover Publications (2003)

  24. Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21(6), 1969–1972 (2000)

    Article  MathSciNet  Google Scholar 

  25. Saad, Y.: ILUT: a dual threshold incomplete LU factorization. Numer. Linear Algebra Appl. 1(4), 387–402 (1994)

    Article  MathSciNet  Google Scholar 

  26. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  MathSciNet  Google Scholar 

  27. Schroder, J.B.: Smoothed aggregation solvers for anisotropic diffusion. Numer. Linear Algebra Appl. 19, 296–312 (2012)

    Article  MathSciNet  Google Scholar 

  28. Silvester, D., Elman, H., Kay, D., Wathen, A.: Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow. J. Comput. Appl. Math. 128(1–2), 261–279 (2001)

    Article  MathSciNet  Google Scholar 

  29. Tang, M., Wang, Y.: An asymptotic preserving method for strongly anisotropic diffusion equations based on field line integration. J. Comput. Phys. 330, 735–748 (2017)

    Article  MathSciNet  Google Scholar 

  30. van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992)

    Article  MathSciNet  Google Scholar 

  31. Wang, Y., Ying, W., Tang, M.: Uniformly convergent scheme for strongly anisotropic diffusion equations with closed field lines. SIAM J. Sci. Comput., pp. B1253–B1276 (2018)

  32. Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)

  33. Yang, C., Claustre, J., Deluzet, F.: Iterative solvers for elliptic problems with arbitrary anisotropy strengths. Multiscale Model. Simul. 16(4), 1795–1823 (2018)

    Article  MathSciNet  Google Scholar 

  34. Yang, C., Deluzet, F., Narski, J.: On the numerical resolution of anisotropic equations with high order differential operators arising in plasma physics. J. Comput. Phys. 386, 502–523 (2019)

    Article  MathSciNet  Google Scholar 

  35. Yang, C., Deluzet, F., Narski, J.: Preserving the accuracy of numerical methods discretizing anisotropic elliptic problems. arXiv:1911.11482

  36. Yavneh, I.: Coarse-grid correction for nonelliptic and singular perturbation problems. SIAM J. Sci. Comput. 19(5), 1682–1699 (1998)

    Article  MathSciNet  Google Scholar 

  37. Yavneh, I., Venner, C.H., Brandt, A.: Fast multigrid solution of the advection problem with closed characteristics. SIAM J. Sci. Comput. 19(1), 111–125 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors acknowledge fruitful discussions with Dr. Fabrice Deluzet from Institut de Mathématiques de Toulouse Université Paul Sabatier. The authors also would like to thank the anonymous reviewers for their valuable comments and suggestions to greatly improve the quality of the paper, and especially for the mention of the work by D. J. Mavriplis and I. Yavneh.

Funding

This work has been supported by the Natural Science Foundation of China under Grant (11901042, 12371432) and the Fundamental Research Funds for the Central Universities (2022FRFK060021). Chang Yang acknowledges support by a public grant from the “Laboratoire d’Excellence Centre International de Mathématiques et d’Informatique” (LabEx CIMI) overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference ANR-11-LABX-0040) in the frame of the PROMETEUS project (PRospect of nOvel nuMerical modEls for elecTric propulsion and low tEmperatUre plaSmas) and from the FrFCM (Fédération de recherche pour la Fusion par Confinement Magnétique) in the frame of the NEMESIA project (Numerical mEthods for Macroscopic models of magnEtized plaSmas and related anIsotropic equAtions).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

    Lingxiao Li

  2. Center for Applied Mathematics of Henan Province, Henan University, Zhengzhou, 450046, China

    Lingxiao Li

  3. School of Mathematics, Harbin Institute of Technology, No. 92 West Dazhi Street, Nangang District, Harbin, 150001, China

    Chang Yang

Authors
  1. Lingxiao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang Yang.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Finite Difference Discretizations

Appendix A: Finite Difference Discretizations

In this part, let us give the finite difference discretizations for the MMAP scheme introduced in Sect. 2.1.

1.1 A.1. Aligned Case

In the aligned case, we consider the magnetic field aligned with z-axis, i.e. \({\textbf{b}}=(0,1)^T\). For the convenience of matrix construction, we write the MMAP scheme of Sect. 2.1 as follows

$$\begin{aligned} \left\{ \begin{array}{ll} - \partial _z^2 q - \partial _x^2\phi = f,&{} (x,z)\in (0,1)\times (0,1),\\ q=0,&{}(x,z)\in \{0,1\}\times [0,1],\\ q=0,&{}(x,z)\in [0,1]\times \{0\}, \\ \partial _z q=0,&{}(x,z)\in [0,1]\times \{1\}, \end{array} \right. \end{aligned}$$
(31)

and

$$\begin{aligned} \left\{ \begin{array}{ll} \varepsilon \partial _z^2 q - \partial _z^2\phi = 0,&{} (x,z)\in (0,1)\times (0,1),\\ \phi =0,&{}(x,z)\in \{0,1\}\times [0,1],\\ \partial _z q=0,&{}(x,z)\in [0,1]\times \{0\},\\ \partial _z\phi = 0,&{} (x,z)\in [0,1]\times \{1\}. \end{array} \right. \end{aligned}$$
(32)

The mesh is defined by

$$\begin{aligned} \left\{ \begin{array}{ll} x_i = (i-1)\Delta x,\ i\in \{1,\dots ,n_x\},\\ z_j = (j-3/2)\Delta z,\ j\in \{1,\dots ,n_z\}, \end{array} \right. \end{aligned}$$

where \(\Delta x = \frac{1}{n_x-1}\) and \(\Delta z = \frac{1}{n_z-2}\). Furthermore, we take \(n_x=N\) and \(n_z=N+1\), so that \(h=\Delta x = \Delta z\). Then a second order finite difference discretization for (31) and (32) can be directly determined as

$$\begin{aligned} \left\{ \begin{array}{ll} \begin{aligned} (-q_{i,j-1} + 2q_{i,j} - q_{i,j+1}) + \\ (-\phi _{i-1,j} + 2\phi _{i,j} - \phi _{i+1,j}) &{} = h^2 f_{i,j}, \end{aligned} &{} (i,j)\in [2,n_x-1]\times [2,n_z-1],\\ q_{1,j}=q_{n_x,j}=0,&{} j\in [1,n_z],\\ q_{i,1}=0,&{} i\in [2,n_x-1],\\ -q_{i,n_z-1} + q_{i,n_z}=0,&{} i\in [2,n_x-1], \end{array} \right. \end{aligned}$$
(33)

and

$$\begin{aligned} \left\{ \begin{array}{ll} \begin{aligned} \varepsilon (q_{i,j-1} - 2q_{i,j} + q_{i,j+1}) + \\ (-\phi _{i,j-1} + 2\phi _{i,j} - &{}\phi _{i,j+1}) = 0, \end{aligned} &{} (i,j)\in [2,n_x-1]\times [2,n_z-1],\\ \phi _{1,j}=\phi _{n_x,j}=0,&{} j\in [1,n_z],\\ -\phi _{i,n_z-1} + \phi _{i,n_z}=0,&{} i\in [2,n_x-1],\\ q_{i,1} - q_{i,2} =0,&{} i\in [2,n_x-1], \end{array} \right. \end{aligned}$$
(34)

where \(\phi _{i,j}\) and \(q_{i,j}\) are numerical approximations of \(\phi (x_i,z_j)\) and \(q(x_i,z_j)\) respectively, and \(f_{i,j}= f(x_i,z_j)\). Now, to construct the linear system to (33)–(34), we define a lexicographic, that is to count first in z direction then in x direction. Denote iline the index of the linear system, we then define a function ’lexico’ to convert from the couple (i, j) to iline as

$$\begin{aligned} iline = \text {lexico}(i,j) = j + (i-1)n_z. \end{aligned}$$
(35)

So the linear system corresponding to (33)–(34) is given by

$$\begin{aligned} \begin{pmatrix} A_{11} &{} A_{12} \\ A_{21} &{} A_{22} \end{pmatrix} \begin{pmatrix} Q \\ \Phi \end{pmatrix} = \begin{pmatrix} F \\ 0 \end{pmatrix}. \end{aligned}$$

At last, in order to get a more symmetrical matrix form, we make some algebraic operations, which yield in the first equation of (33) as

$$\begin{aligned} \begin{array}{llll} -\phi _{i-1,j} + 2\phi _{i,j} - \phi _{i+1,j} &{} \rightarrow &{} 2\phi _{i,j} - \phi _{i+1,j}, &{} (i,j)\in \{2\}\times [2,n_z-1],\\ -\phi _{i-1,j} + 2\phi _{i,j} - \phi _{i+1,j} &{} \rightarrow &{} -\phi _{i-1,j} + 2\phi _{i,j},&{} (i,j)\in \{n_x-1\}\times [2,n_z-1],\\ -q_{i,j-1} + 2q_{i,j} - q_{i,j+1} &{} \rightarrow &{} 2q_{i,j} - q_{i,j+1}, &{} (i,j)\in [2,n_x-1]\times \{2\}. \end{array} \end{aligned}$$

Thanks to these algebraic operations, the matrices \(A_{11}\) and \(A_{12}\) become symmetric.

1.2 A.2. Misaligned Case

Now let us concentrate on discretization for the general MMAP scheme (8)–(9). Again, this time we will consider a second order finite difference method.

Firstly, the mesh is defined different from the previous subsection, since we have to discretize cross derivative on all interior grids. For this, the mesh is defined by

$$\begin{aligned} \left\{ \begin{array}{ll} x_i = (i-1)\Delta x,\ i\in \{1,\dots ,n_x\},\\ z_j = (j-1)\Delta z,\ j\in \{1,\dots ,n_z\}, \end{array} \right. \end{aligned}$$

where \(\Delta x = \frac{1}{n_x-1}\) and \(\Delta z = \frac{1}{n_z-1}\). Again, we take \(n_x=n_z=N\), so that \(h=\Delta x = \Delta z\).

Secondly, we notice that

$$\begin{aligned} \Delta \phi = \Delta _\bot \phi + \Delta _\Vert \phi , \end{aligned}$$

its corresponding discretization on interior grids is

$$\begin{aligned} \Delta ^h \phi _{i,j} = \Delta ^h_\bot \phi _{i,j} + \Delta ^h_\Vert \phi _{i,j},\quad (i,j)\in \{2,\dots ,n_x-1\}\times \{2,\dots ,n_z-1\}. \end{aligned}$$

Using the usual central discretization, \(\Delta \phi \) can be approximated as

$$\begin{aligned} h^2\Delta ^h \phi _{i,j} = \phi _{i+1,j} + \phi _{i-1,j} + \phi _{i,j+1} + \phi _{i,j-1} - 4 \phi _{i,j}. \end{aligned}$$

Then, let us denote \({\textbf{b}}\otimes {\textbf{b}}\) by

$$\begin{aligned} {\textbf{b}}\otimes {\textbf{b}} = \begin{pmatrix} b^{11} &{} b^{12} \\ b^{21} &{} b^{22} \end{pmatrix}, \end{aligned}$$

thus we can recast \(\Delta _\Vert \phi \) as

$$\begin{aligned} \Delta _\Vert \phi = \partial _x G + \partial _z H, \end{aligned}$$

where \(G = b^{11}\partial _x\phi + b^{12}\partial _z\phi \), \(H = b^{21}\partial _x\phi + b^{22}\partial _z\phi \). Similarly, we get approximate \(\Delta _\Vert \phi \) as

$$\begin{aligned} h^2\Delta ^h_\Vert \phi _{i,j} = (G^h_{i+1/2,j} - G^h_{i-1/2,j}) + (H^h_{i,j+1/2} - H^h_{i,j-1/2}), \end{aligned}$$

where

$$\begin{aligned} G^h_{i+1/2,j}&= b^{11}_{i+1/2,j}(\phi _{i+1,j} - \phi _{i,j}) + \frac{1}{4}b^{12}_{i+1/2,j}(\phi _{i+1,j+1} + \phi _{i,j+1} - \phi _{i+1,j-1} - \phi _{i,j-1}), \\ H^h_{i,j+1/2}&= \frac{1}{4}b^{21}_{i+1/2,j}(\phi _{i+1,j+1} + \phi _{i+1,j} - \phi _{i-1,j+1} - \phi _{i-1,j}) + b^{22}_{i,j+1/2}(\phi _{i,j+1} - \phi _{i,j}). \end{aligned}$$

At last, we obtain immediately approximated \(\Delta _\bot \phi \) by

$$\begin{aligned} \Delta ^h_\bot \phi _{i,j} = \Delta ^h \phi _{i,j} - \Delta ^h_\Vert \phi _{i,j}. \end{aligned}$$

Thanks to these discrete operators, the sub-blocks \(A_{22}\), \(A_{21}\), \(A_{11}\) and \(A_{12}\) from the linear system (10) correspond to \(-\Delta ^h_\Vert \phi \), \(\varepsilon \Delta ^h_\Vert q\), \(-\Delta ^h_\Vert q\) and \(-\Delta ^h_\bot \phi \) respectively on interior grids. Finally, let us consider discretization for boundary conditions. From (8)–(9), we see that it is necessary to approximate \({\textbf{n}}\cdot \nabla _\bot \phi \) and \({\textbf{n}}\cdot \nabla _\Vert q\) on \([0,1]\times \{0,1\}\). We only present the approximation on boundary \([0,1]\times \{0\}\), the one on boundary \([0,1]\times \{1\}\) is similar. Notice that the normal direction on boundary \([0,1]\times \{0\}\) is \({\textbf{n}}=(0,1)^T\), we thus has

$$\begin{aligned} {\textbf{n}}\cdot \nabla _\bot \phi&= b^{11} \partial _z \phi - b^{12}\partial _x \phi ,\\ {\textbf{n}}\cdot \nabla _\Vert q&= b^{21} \partial _x q + b^{22}\partial _z q. \end{aligned}$$

Now applying center discretizations, we get a second approximation for above differential operations

$$\begin{aligned} h {\textbf{n}}\cdot \nabla ^h_\bot \phi _{i,1}&= \frac{1}{2}b^{11}_{i,1}(-3\phi _{i,1} + 4\phi _{i,2} - \phi _{i,3}) - \frac{1}{2}b^{12}_{i,1}(\phi _{i+1,1} - \phi _{i-1,1}), \\ h {\textbf{n}}\cdot \nabla ^h_\Vert q_{i,1}&= \frac{1}{2}b^{21}_{i,1}(q_{i+1,1} - q_{i-1,1}) + \frac{1}{2}b^{22}_{i,1}(-3q_{i,1} + 4q_{i,2} - q_{i,3}). \end{aligned}$$

At last, substituting \({\textbf{n}}\cdot \nabla _\bot \phi \) and \({\textbf{n}}\cdot \nabla _\Vert q\) in (8)–(9) by the above approximation, we obtain desired second order finite difference schemes.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, L., Yang, C. Block Preconditioning Methods for Asymptotic Preserving Scheme Arising in Anisotropic Elliptic Problems. J Sci Comput 99, 63 (2024). https://doi.org/10.1007/s10915-024-02524-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-024-02524-2

Keywords

Mathematics Subject Classification