[A2] Numerical Analysis for PDEs and Nonlocal Models

I am interested in designing and analyzing finite difference methods, finite element methods (Galerkin, Petrov-Galerkin, DG) and meshfree methods for approximating PDEs and nonlocal models arising in science and engineering applications.

Numerical analysis for PDEs

  1. Q. Ye and X. Tian, Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation, preprint, 2022. [Interactive Website Examples]

  2. L. Demkowicz, T. Führer, N. Heuer, and X. Tian, The Double Adaptivity Paradigm (How to circumvent the discrete inf-sup conditions of Babuška and Brezzi), Comput. Math. with Appl., 95, 41-66, 2021.

Numerical analysis for nonlocal and fractional models

  1. M. Pasetto, Z. Shen, M. D’Elia, X. Tian, N.Trask and D. Kamensky, Efficient optimization-based quadrature for variational discretization of nonlocal problems, Comput. Methods Appl. Mech. Eng.,396, 115104, 2022.

  2. Y. Leng, X. Tian, L. Demkowicz, H. Gomez and J. Foster, A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems, J. Comput. Phys., 425, 110919, 2021.

  3. Y. Leng, X. Tian, N. Trask and J. Foster, Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion, SIAM J. Numer. Anal., 59(1), 88-118, 2021.

  4. M. D’Elia, Q. Du, C. Glusa, M. Gunzburger, X. Tian, and Z. Zhou, Numerical methods for nonlocal and fractional models, Acta Numerica, 29, 1-124, 2020.

  5. Y. Leng, X. Tian, N. Trask and J. Foster, Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation, Comput. Methods Appl. Mech. Eng., 370, 113264, 2020.

  6. X. Tian and Q. Du, Asymptotically compatible schemes for robust discretization of parametrized problems with applications to nonlocal models, SIAM Rev., 62(1), 199–227, 2020 (SIAM Review SIGEST Award).

  7. Q. Du, L. Ju, J. Lu and X. Tian A discontinuous Galerkin method with penalty for one-dimensional nonlocal diffusion problems, Comm. Appl. Math. Comput., 2(1):31-55, 2020.

  8. X. Tian and B. Engquist, Fast algorithm for computing nonlocal operators with finite interaction distance, Comm. Math. Sci., 17(6), 1653 – 1670, 2019.

  9. Q. Du, Y. Tao, X. Tian and J. Yang, Asymptotically compatible numerical approximations of multidimensional nonlocal diffusion models and nonlocal Green's functions, IMA J. Numer. Anal., 39(2),607–625, 2018.

  10. Q. Du, Y. Tao, X. Tian , and J. Yang, Robust a posteriori stress analysis for approximations of nonlocal models via nonlocal gradients, Comput. Methods Appl. Mech. Eng., 310:605-627, 2016.

  11. X. Tian , Q. Du and M. Gunzburger, Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains, Advances in Comp. Math., 42(6):1363-1380, 2016.

  12. X. Tian and Q. Du, Nonconforming discontinuous Galerkin methods for nonlocal varational problems, SIAM J. Numer. Anal., 53 (2015), 762-781.

  13. X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM J. Numer. Anal. 52 (2014), 1641-1665.

  14. X. Tian and Q. Du, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J. Numer. Anal. 51 (2013), 3458-3482.