Publications

Journal Papers (boldface name means the corresponding author):

1. G. Wang, M. Cai, X. Dong, and Y. He, A virtual element method for the four-field poroelasticity problem on polygonal meshes and its simulation in brain edema, NMPDE, accepted.
2. M. Cai, M. Kuchta, J. Li, Z. Li, and K. Mardal, Parameter-robust preconditioners for a four-field thermo-poroelasticity model, SIAM J. Sci. Comput., accepted.
3. O. Olabanjo, E. Aigbokhan, E. Akor, D. Kaczka, and M. Cai, Finite element simulation of lung parenchyma deformation based on porcine data, Computer Methods in Biomechanics and Biomedical Engineering, 2025, doi: https://doi.org/10.1080/10255842.2025.2556314.
4. W. Duan, X. Shen, P. Piersanti, Y. Liu, and M. Cai, An efficient decoupling algorithm for thermoelastic dynamic system of elliptic membrane shell, Int. J. Eng. Sci., https://doi.org/10.1016/j.ijengsci.2025.104353
5. M. Cai, J. Li, Z. Li, and Q. Liu, An efficient iterative decoupling method for thermo-poroelasticity based on a four-field formulation, Computers & Mathematics with Applications, Volume 195, 1 October 2025, Pages 139-160. https://www.sciencedirect.com/science/article/abs/pii/S0898122125003128
6. B. Han, E. Akor, A. Cruz, M. Cai, and D. Kaczka, Simulation of central airway gas flow dynamics during conventional and multifrequency ventilation, Journal of Biomechanical Engineering, ASME, Oct 2025, 147(10): 101004. https://doi.org/10.1115/1.4069032
7. J. Zhao, H. Chen, M. Cai, and S. Sun, An optimally convergent parallel splitting algorithm for the multiple-network poroelasticity model, J. Comput. Phys., Volume 539, 15 October 2025, 114214. https://doi.org/10.1016/j.jcp.2025.114214
8. E. A. Akor, B. Han, M. Cai, C. Lin, and D. W. Kaczk, Forward computational modeling of respiratory airflow, Appl. Sci. 2024, 14(24), 11591; doi: https://doi.org/10.3390/app142411591.
9. H. Gu, M. Cai, and J. Li, Convergence analysis of a global-in-time iterative decoupled algorithm for Biot’s model, submitted to Advances in Applied Mathematics and Mechanics, accepted. doi: 10.4208/aamm.OA-2024-0074
10. Y. Zeng, L. Zhong, F. Wang, M. Cai, and S. Zhang, Low regularity error analysis for an H-(div)-conforming discontinuous Galerkin approximation of Stokes problem, J. Comput. Appl. Math., https://doi.org/10.1016/j.cam.2024.116118
11. Y. Zeng, L. Zhong, M. Cai, F. Wang, and S. Zhang, Conforming and nonconforming virtual element methods for Signorini problems, J. Sci. Comput., doi: 10.1007/s10915-024-02562-w.
12. M. Cai, G. Ju, and J. Li, Schur complement based preconditioners for two-fold saddle point and block tridiagonal problems, Communications in Mathematical Research, Vol. 40, No. 2, pp. 214-244, doi: 10.4208/cmr.2023-0051.
13. Y. Zeng, L. Zhong, F. Wang, S. Zhang, and M. Cai, A pressure-robust numerical scheme for the Stokes equations based on the WOPSIP DG approach, J. Comput. Appl. Math., Vol 445, (2024), 115819. https://doi.org/10.1016/j.cam.2024.115819
14. M. Cai, H. Gu, J. Li, and M. Mu, Some optimally convergent algorithms for decoupling the computation of Biot’s model, J. Sci. Comput., 97 (2023), no. 2, Paper No. 48, 33 , doi:10.1007/s10915-023-02365-5 .
15. M. Cai, H. Gu, P. Hong, and J. Li, A combination of physics-informed neural networks with the fixed-stress splitting iteration for solving Biot's model, Front. Appl. Math. Stat., Sec. Mathematics of Computation and Data Science, Volume 9 - 2023, doi: 10.3389/fams.2023.1206500.
16. H. Gu, M.Cai, J. Li, and G.Ju, A priori error estimates of two monolithic schemes for Biot’s consolidation model, Numer. Methods Partial Differ. Eq. (2023),1–22. https://doi.org/10.1002/num.23059 (code available upon request).
17. Y. Zeng, M. Cai, and L. Zhong, A coupled method combining Crouzeix-Raviart nonconforming elements with continuous linear elements for Biot consolidation model, J. Comput. Math., J. Comput. Math. 42 (2024), no. 4, 911–931, doi:10.4208/jcm.2212-m2021-0231. (code available upon request).
18. H. Gu, M. Cai, and J. Li, An iterative decoupled algorithm with unconditional stability for Biot model, Math. Comput., Vol. 92, No. 341, May 2023, pp. 1087–1108. doi
19. F. Wang, M. Cai, G. Wang, and Y. Zeng, A mixed virtual element method for Biot’s consolidation model, Computers & Mathematics with Applications, Vol 126, 15 November 2022, pp. 31-42. doi: https://www.sciencedirect.com/science/article/pii/S0898122122003789
20. L. Zhang, M. Cai, and M. Mu, Decoupling PDE computation with instrinsic or inertial Robin interface condition, Electronic Research Archive, June 2021, Volume 29, Issue 2, pp. 2007-2028.
21. G. Ju, M. Cai and J. Li, and J. Tian, Parameter-robust multiphysics algorithms for Biot model: with application in brain swelling simulation, Mathematics and Computers in Simulation, Vol 177, Nov. 2020, 385-403.
22. J. Tian, M. Cai, Z. Luo, Y. You, and G. Chen, A 3D OpenFOAM based Finite Volume solver for incompressible Oldroyd-B model with infinity relaxation time. Commun. Nonlinear Sci. Numer. Simul. 78 (2019),04876.
23. L. Zhang, M. Cai and M. Mu, A multirate approach for fluid-structure interaction computation with decoupled methods, Commun. Comput. Phys., 2020, 27(4):1014-1031.
24. G. Zhang and M. Cai, Normal mode analysis of 3D incompressible viscous fluid flow models, Appl. Anal. Vol. 100, 2021 - Issue 1, pp. 116-134. doi.
25. Y. Zeng, M. Cai and F. Wang, An H(div)-conforming finite element method for Biot’s consolidation model, East Asian J. Appl. Math. , Vol. 9, No. 3, 2019, pp. 558-579. doi.
26. M. Cai and G. Zhang, Comparisons of some iterative algorithms for Biot equations, (a special issue "Differential Equations, Almost Periodicity, and Almost Automorphy”, dedicated to the memory of Prof. V.V. Zhikov.). Int. J. Evol. Equ., Vol. 10, No. 3-4, 2017, pp. 267–282. doi.
27. M. Cai, P. Huang, and M. Mu, Some multilevel decoupled algorithms for a mixed Navier-Stokes/Darcy model. Adv. Comput. Math. 44 (2018), no. 1, 115–145. doi.
28. F. Qin, J. Chen, Z. Li and M. Cai. A Cartesian grid nonconforming immersed finite element method for planar elasticity interface problems. Comput. Math. Appl., 73, no.3, 2017, pp. 404-418. doi.
29. P. Huang, J. Chen and M. Cai. A mortar method using nonconforming and mixed finite elements for the coupled Stokes-Darcy model. Adv. Appl. Math. Mech., 9(3), 2017, pp. 596-620. doi
30. M. Cai and L. Pavarino, Hybrid and multiplicative overlapping Schwarz algorithms with standard coarse space for mixed linear elasticity and Stokes problems. Commun. Comput. Phys., 20(4), 2016, pp. 989-1015.doi
31. P. Huang, M. Cai and F. Wang, A Newton type linearization based two grid method for coupling fluid flow with porous media flow, Appl. Numer. Math., 106, 2016, pp. 182-198. doi.
32. M. Cai, L. Pavarino and O. Widlund, Overlapping Schwarz methods with a standard coarse space for almost incompressible linear elasticity. SIAM J. Sci. Comput., 37(2), pp. A811-A831.doi
33. M. Cai, Analysis of some projection method based preconditioners for models of incompressible flow, Appl. Numer. Math., 90, 2015, pp. 77-90.doi
34. M. Cai, A. Nonaka, B. Griffith, J. Bell and A. Donev, Efficient variable-coefficient finite volume Stokes solvers, Commmun. Comput. Phys., 16 no. 5, 2014, pp. 1263-1297.doi
35. H. Liu, M. Cai, C. Yang, J. Zheng, R. Bach, M. Kural, K. Billiar, D. Muccigrosso, D. Lu, D. Tang, IVUS-based computational modeling and planar biaxial artery material properties for human coronary plaque vulnerability assessment, Mol Cell Biomech. 2012 Mar;9(1) pp.77-93. doi
36. P. Huang, J. Chen and M. Cai, A mixed and nonconforming FEM with nonmatching meshes for a coupled Stokes-Darcy model, J. Sci. Comput. 53 (2012), no. 2, pp. 377–394. doii
37. M. Kural, M. Cai, D. Tang, T. Gwyther, J. Zheng and K. Billiar, Planar biaxial characterization of diseased humen coronary and carotid arteries for computational modeling, J. of Biomechanics, Vol. 45, Iss. 5, pp. 790–798 doi
38. M. Cai and M. Mu, A multilevel decoupled method for a mixed Stokes/Darcy model, J. Comput. Appl. Math., 236 (2012), no. 9, pp. 2452–2465. doi
39. M. Cai, M. Mu and J. Xu, Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach, SIAM J. Numer. Anal., Vol. 47 (2009), pp. 3325-3338. doi (Top 20 Most Read Articles, Oct. 2009 - Jan. 2010)
40. M. Cai, M. Mu and J. Xu, Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications, J. Comput. Appl. Math., Vol. 233 (2009), pp. 346-355. doi
41. M. Cai, X. Jin and Y. Wei, A generalization of T. Chan's preconditioner, Lin. Alg. Appl., Vol. 407 (2005), pp. 11-18. doi
42. M. Cai and X. Jin, BCCB preconditioners for solving linear systems from delay differential equations, Comput. Math. Appl. Vol. 50 (2005), pp. 281-288. doi
43. M. Cai and X. Jin, A note on T. Chan's precondtioner, Lin. Alg. Appl., Vol. 376 (2004), pp. 283-290. doi

Draft in revision and preparation:

Overlapping Schwarz algorithms for Biot's model.

Book Chapters:

M. Cai, Decoupled algorithms for the coupled surface/subsurface flow interaction problems, E-Book Series Progress in Computational Physics (PiCP), Volume 2. doi
M. Cai, M. Mu, and L. Zhang, Decoupling Techniques for Coupled PDE Models in Fluid Dynamics, Book: Advances in Fusion Energy Research - From Theory to Models, Algorithms, and Applications, edited by: Dr. Bruno Carpentieri and Dr. Aamir Shahzad, 2022, ISBN 978-1-80355-415-0. doi

Thesis Works:

Modeling and numerical simulation for the coupling of surface flow with subsurface flow, M. Cai, Ph. D. thesis. doi
Some properties of T. Chan's preconditioner and applications in numerical differential equations, M. Cai, Master thesis. doi

Selected Conference Papers, Tech. Reports, Proceedings, and Posters:

M. Cai, C. Yang, M. H. Kural, R. Bach, D. Muccigrosso, D. Yang, J. Zheng, C. L. Billiar and D. Tang, Intravascular ultrasound (IVUS)-based computational modeling and planar biaxial artery material properties for human coronary plaque vulnerability assessment, conference paper, International Conference on Computational and Experimental Engineering and Sciences. ICCES, vol.19, no.4, pp.97-104, 2011. doi
M. Cai, C. Yang, P. Yu, B. Nguyen, M. Tao, C. K. Ozaki and D. Tang, Intimal hyperplasia and total vessel thickness correlate positively with flow shear stress in a murine vein graft model with advanced focal stenosis, ATVB 2011 Poster.
M. Cai and S. Turek, Numerical studies of time dependent Ginzburg-Landau model by FEM with moving grid deformation, (Tech. Report. Techn. Univ., Fak. fur Mathematik, 2010.)

Publications

Journal Papers (boldface name means the corresponding author):

1. G. Wang, M. Cai, X. Dong, and Y. He, A virtual element method for the four-field poroelasticity problem on polygonal meshes and its simulation in brain edema, NMPDE, accepted.
2. M. Cai, M. Kuchta, J. Li, Z. Li, and K. Mardal, Parameter-robust preconditioners for a four-field thermo-poroelasticity model, SIAM J. Sci. Comput., accepted.
3. O. Olabanjo, E. Aigbokhan, E. Akor, D. Kaczka, and M. Cai, Finite element simulation of lung parenchyma deformation based on porcine data, Computer Methods in Biomechanics and Biomedical Engineering, 2025, doi: https://doi.org/10.1080/10255842.2025.2556314.
4. W. Duan, X. Shen, P. Piersanti, Y. Liu, and M. Cai, An efficient decoupling algorithm for thermoelastic dynamic system of elliptic membrane shell, Int. J. Eng. Sci., https://doi.org/10.1016/j.ijengsci.2025.104353
5. M. Cai, J. Li, Z. Li, and Q. Liu, An efficient iterative decoupling method for thermo-poroelasticity based on a four-field formulation, Computers & Mathematics with Applications, Volume 195, 1 October 2025, Pages 139-160. https://www.sciencedirect.com/science/article/abs/pii/S0898122125003128
6. B. Han, E. Akor, A. Cruz, M. Cai, and D. Kaczka, Simulation of central airway gas flow dynamics during conventional and multifrequency ventilation, Journal of Biomechanical Engineering, ASME, Oct 2025, 147(10): 101004. https://doi.org/10.1115/1.4069032
7. J. Zhao, H. Chen, M. Cai, and S. Sun, An optimally convergent parallel splitting algorithm for the multiple-network poroelasticity model, J. Comput. Phys., Volume 539, 15 October 2025, 114214. https://doi.org/10.1016/j.jcp.2025.114214
8. E. A. Akor, B. Han, M. Cai, C. Lin, and D. W. Kaczk, Forward computational modeling of respiratory airflow, Appl. Sci. 2024, 14(24), 11591; doi: https://doi.org/10.3390/app142411591.
9. H. Gu, M. Cai, and J. Li, Convergence analysis of a global-in-time iterative decoupled algorithm for Biot’s model, submitted to Advances in Applied Mathematics and Mechanics, accepted. doi: 10.4208/aamm.OA-2024-0074
10. Y. Zeng, L. Zhong, F. Wang, M. Cai, and S. Zhang, Low regularity error analysis for an H-(div)-conforming discontinuous Galerkin approximation of Stokes problem, J. Comput. Appl. Math., https://doi.org/10.1016/j.cam.2024.116118
11. Y. Zeng, L. Zhong, M. Cai, F. Wang, and S. Zhang, Conforming and nonconforming virtual element methods for Signorini problems, J. Sci. Comput., doi: 10.1007/s10915-024-02562-w.
12. M. Cai, G. Ju, and J. Li, Schur complement based preconditioners for two-fold saddle point and block tridiagonal problems, Communications in Mathematical Research, Vol. 40, No. 2, pp. 214-244, doi: 10.4208/cmr.2023-0051.
13. Y. Zeng, L. Zhong, F. Wang, S. Zhang, and M. Cai, A pressure-robust numerical scheme for the Stokes equations based on the WOPSIP DG approach, J. Comput. Appl. Math., Vol 445, (2024), 115819. https://doi.org/10.1016/j.cam.2024.115819
14. M. Cai, H. Gu, J. Li, and M. Mu, Some optimally convergent algorithms for decoupling the computation of Biot’s model, J. Sci. Comput., 97 (2023), no. 2, Paper No. 48, 33 , doi:10.1007/s10915-023-02365-5 .
15. M. Cai, H. Gu, P. Hong, and J. Li, A combination of physics-informed neural networks with the fixed-stress splitting iteration for solving Biot's model, Front. Appl. Math. Stat., Sec. Mathematics of Computation and Data Science, Volume 9 - 2023, doi: 10.3389/fams.2023.1206500.
16. H. Gu, M.Cai, J. Li, and G.Ju, A priori error estimates of two monolithic schemes for Biot’s consolidation model, Numer. Methods Partial Differ. Eq. (2023),1–22. https://doi.org/10.1002/num.23059 (code available upon request).
17. Y. Zeng, M. Cai, and L. Zhong, A coupled method combining Crouzeix-Raviart nonconforming elements with continuous linear elements for Biot consolidation model, J. Comput. Math., J. Comput. Math. 42 (2024), no. 4, 911–931, doi:10.4208/jcm.2212-m2021-0231. (code available upon request).
18. H. Gu, M. Cai, and J. Li, An iterative decoupled algorithm with unconditional stability for Biot model, Math. Comput., Vol. 92, No. 341, May 2023, pp. 1087–1108. doi
19. F. Wang, M. Cai, G. Wang, and Y. Zeng, A mixed virtual element method for Biot’s consolidation model, Computers & Mathematics with Applications, Vol 126, 15 November 2022, pp. 31-42. doi: https://www.sciencedirect.com/science/article/pii/S0898122122003789
20. L. Zhang, M. Cai, and M. Mu, Decoupling PDE computation with instrinsic or inertial Robin interface condition, Electronic Research Archive, June 2021, Volume 29, Issue 2, pp. 2007-2028.
21. G. Ju, M. Cai and J. Li, and J. Tian, Parameter-robust multiphysics algorithms for Biot model: with application in brain swelling simulation, Mathematics and Computers in Simulation, Vol 177, Nov. 2020, 385-403.
22. J. Tian, M. Cai, Z. Luo, Y. You, and G. Chen, A 3D OpenFOAM based Finite Volume solver for incompressible Oldroyd-B model with infinity relaxation time. Commun. Nonlinear Sci. Numer. Simul. 78 (2019),04876.
23. L. Zhang, M. Cai and M. Mu, A multirate approach for fluid-structure interaction computation with decoupled methods, Commun. Comput. Phys., 2020, 27(4):1014-1031.
24. G. Zhang and M. Cai, Normal mode analysis of 3D incompressible viscous fluid flow models, Appl. Anal. Vol. 100, 2021 - Issue 1, pp. 116-134. doi.
25. Y. Zeng, M. Cai and F. Wang, An H(div)-conforming finite element method for Biot’s consolidation model, East Asian J. Appl. Math. , Vol. 9, No. 3, 2019, pp. 558-579. doi.
26. M. Cai and G. Zhang, Comparisons of some iterative algorithms for Biot equations, (a special issue "Differential Equations, Almost Periodicity, and Almost Automorphy”, dedicated to the memory of Prof. V.V. Zhikov.). Int. J. Evol. Equ., Vol. 10, No. 3-4, 2017, pp. 267–282. doi.
27. M. Cai, P. Huang, and M. Mu, Some multilevel decoupled algorithms for a mixed Navier-Stokes/Darcy model. Adv. Comput. Math. 44 (2018), no. 1, 115–145. doi.
28. F. Qin, J. Chen, Z. Li and M. Cai. A Cartesian grid nonconforming immersed finite element method for planar elasticity interface problems. Comput. Math. Appl., 73, no.3, 2017, pp. 404-418. doi.
29. P. Huang, J. Chen and M. Cai. A mortar method using nonconforming and mixed finite elements for the coupled Stokes-Darcy model. Adv. Appl. Math. Mech., 9(3), 2017, pp. 596-620. doi
30. M. Cai and L. Pavarino, Hybrid and multiplicative overlapping Schwarz algorithms with standard coarse space for mixed linear elasticity and Stokes problems. Commun. Comput. Phys., 20(4), 2016, pp. 989-1015.doi
31. P. Huang, M. Cai and F. Wang, A Newton type linearization based two grid method for coupling fluid flow with porous media flow, Appl. Numer. Math., 106, 2016, pp. 182-198. doi.
32. M. Cai, L. Pavarino and O. Widlund, Overlapping Schwarz methods with a standard coarse space for almost incompressible linear elasticity. SIAM J. Sci. Comput., 37(2), pp. A811-A831.doi
33. M. Cai, Analysis of some projection method based preconditioners for models of incompressible flow, Appl. Numer. Math., 90, 2015, pp. 77-90.doi
34. M. Cai, A. Nonaka, B. Griffith, J. Bell and A. Donev, Efficient variable-coefficient finite volume Stokes solvers, Commmun. Comput. Phys., 16 no. 5, 2014, pp. 1263-1297.doi
35. H. Liu, M. Cai, C. Yang, J. Zheng, R. Bach, M. Kural, K. Billiar, D. Muccigrosso, D. Lu, D. Tang, IVUS-based computational modeling and planar biaxial artery material properties for human coronary plaque vulnerability assessment, Mol Cell Biomech. 2012 Mar;9(1) pp.77-93. doi
36. P. Huang, J. Chen and M. Cai, A mixed and nonconforming FEM with nonmatching meshes for a coupled Stokes-Darcy model, J. Sci. Comput. 53 (2012), no. 2, pp. 377–394. doii
37. M. Kural, M. Cai, D. Tang, T. Gwyther, J. Zheng and K. Billiar, Planar biaxial characterization of diseased humen coronary and carotid arteries for computational modeling, J. of Biomechanics, Vol. 45, Iss. 5, pp. 790–798 doi
38. M. Cai and M. Mu, A multilevel decoupled method for a mixed Stokes/Darcy model, J. Comput. Appl. Math., 236 (2012), no. 9, pp. 2452–2465. doi
39. M. Cai, M. Mu and J. Xu, Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach, SIAM J. Numer. Anal., Vol. 47 (2009), pp. 3325-3338. doi (Top 20 Most Read Articles, Oct. 2009 - Jan. 2010)
40. M. Cai, M. Mu and J. Xu, Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications, J. Comput. Appl. Math., Vol. 233 (2009), pp. 346-355. doi
41. M. Cai, X. Jin and Y. Wei, A generalization of T. Chan's preconditioner, Lin. Alg. Appl., Vol. 407 (2005), pp. 11-18. doi
42. M. Cai and X. Jin, BCCB preconditioners for solving linear systems from delay differential equations, Comput. Math. Appl. Vol. 50 (2005), pp. 281-288. doi
43. M. Cai and X. Jin, A note on T. Chan's precondtioner, Lin. Alg. Appl., Vol. 376 (2004), pp. 283-290. doi

Draft in revision and preparation:

Overlapping Schwarz algorithms for Biot's model.

Book Chapters:

M. Cai, Decoupled algorithms for the coupled surface/subsurface flow interaction problems, E-Book Series Progress in Computational Physics (PiCP), Volume 2. doi
M. Cai, M. Mu, and L. Zhang, Decoupling Techniques for Coupled PDE Models in Fluid Dynamics, Book: Advances in Fusion Energy Research - From Theory to Models, Algorithms, and Applications, edited by: Dr. Bruno Carpentieri and Dr. Aamir Shahzad, 2022, ISBN 978-1-80355-415-0. doi

Thesis Works:

Modeling and numerical simulation for the coupling of surface flow with subsurface flow, M. Cai, Ph. D. thesis. doi
Some properties of T. Chan's preconditioner and applications in numerical differential equations, M. Cai, Master thesis. doi

Selected Conference Papers, Tech. Reports, Proceedings, and Posters:

M. Cai, C. Yang, M. H. Kural, R. Bach, D. Muccigrosso, D. Yang, J. Zheng, C. L. Billiar and D. Tang, Intravascular ultrasound (IVUS)-based computational modeling and planar biaxial artery material properties for human coronary plaque vulnerability assessment, conference paper, International Conference on Computational and Experimental Engineering and Sciences. ICCES, vol.19, no.4, pp.97-104, 2011. doi
M. Cai, C. Yang, P. Yu, B. Nguyen, M. Tao, C. K. Ozaki and D. Tang, Intimal hyperplasia and total vessel thickness correlate positively with flow shear stress in a murine vein graft model with advanced focal stenosis, ATVB 2011 Poster.
M. Cai and S. Turek, Numerical studies of time dependent Ginzburg-Landau model by FEM with moving grid deformation, (Tech. Report. Techn. Univ., Fak. fur Mathematik, 2010.)

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