Geodesic
Two-grid methods for a new mixed finite element approximation of semilinear parabolic integro-differential equations
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 2, pp. 167-185.

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In this paper, we present a two-grid scheme for a semilinear parabolic integro-differential equation using a new mixed finite element method. The gradient for the method belongs to the space of square integrable functions instead of the classical $H(\mathrm{div};\Omega)$ space. The velocity and the pressure are approximated by a $P_0^2-P_1$ pair which satisfies an inf-sup condition. Firstly, we solve the original nonlinear problem on the coarse grid in our two-grid scheme. Then, to linearize the discretized equations, we use Newton’s iteration on the fine grid twice. It is shown that the algorithm can achieve an asymptotically optimal approximation as long as the mesh sizes satisfy $h=\mathcal{O}(H^6|\ln H|^2)$. As a result, solving such a large class of nonlinear equations will not be much more difficult than solving one linearized equation. Finally, a numerical experiment is provided to verify the theoretical results of the two-grid method. 
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C. Liu; T. Hou. Two-grid methods for a new mixed finite element approximation of semilinear parabolic integro-differential equations. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 2, pp. 167-185. http://geodesic.mathdoc.fr/item/SJVM_2019_22_2_a4/

[1] Brezzi F., Fortin M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991 | MR | Zbl

[2] Bi C., Ginting V., “Two-grid finite volume element method for linear and nonlinear elliptic problems”, Numer. Math., 108:2 (2007), 177-198 | DOI | MR | Zbl

[3] Bi C., Ginting V., “Two-grid discontinuous Galerkin method for quasi-linear elliptic problems”, J. Sci. Comput., 49:3 (2011), 311-331 | DOI | MR | Zbl

[4] Chen L., Chen Y., “Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods”, J. Sci. Comput., 49:3 (2011), 383-401 | DOI | MR | Zbl

[5] Chen Y., Huang Y., Yu D., “A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations”, Int. J. Numer. Meth. Eng., 57:2 (2003), 193-209 | DOI | MR | Zbl

[6] Chen Y., Liu H., Liu S., “Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods”, Int. J. Numer. Meth. Eng., 69 (2007), 408-422 | DOI | MR | Zbl

[7] Cannon J.R., Lin Y., “A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory”, SIAM J. Numer. Anal., 27 (1990), 595-607 | DOI | MR | Zbl

[8] Chen S.C., Chen H.R., “New mixed element schemes for a second-order elliptic problem”, Mathematica Numerica Sinica, 32:2 (2010), 213-218 | MR | Zbl

[9] Ciarlet P.G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978 | MR | Zbl

[10] Dawson C., Wheeler M.F., Woodward C.S., “A two-grid finite difference scheme for nonlinear parabolic equations”, SIAM J. Numer. Anal., 35 (1998), 435-452 | DOI | MR | Zbl

[11] Dawson C.N., Wheeler M.F., “Two-grid methods for mixed finite element approximations of nonlinear parabolic equations”, Contemp. Maths., 180, 1994, 191-203 | DOI | MR | Zbl

[12] Douglas J., Roberts J.E., “Global estimates for mixed methods for second order elliptic equations”, Math. Comp., 44 (1985), 39-52 | DOI | MR | Zbl

[13] Ewing R.E., Lin Y.P., Sun T., Wang J.P., Zhang S.H., “Sharp $L^2$-error estimates and superconvergence of mixed finite element methods for non-Fickian flows in porous media”, SIAM J. Numer. Anal., 40:4 (2002), 1538-1560 | DOI | MR | Zbl

[14] Yanik E.G., Fairweather G., “Finite element methods for parabolic and hyperbolic partial integro-differential equations”, Nonlinear Anal., 12 (1988), 785-809 | DOI | MR | Zbl

[15] Grisvard P., Elliptic Problems in Nonsmooth Domains, Pitman Advanced Pub., Boston, 1985 | MR | Zbl

[16] Huang Y., Shi Z.H., Tang T., Xue W., “A multilevel successive iteration method for nonlinear elliptic problems”, Math. Comput., 73:246 (2004), 525-539 | DOI | MR | Zbl

[17] Huang Y., Xue W., “Convergence of finite element approximations and multilevel linearization for Ginzburg-Landau model of d-wave superconductors”, Adv. Comput. Math., 17 (2002), 309-330 | DOI | MR | Zbl

[18] Lin Y., “Galerkin methods for nonlinear parabolic integrodifferential equations with nonlinear boundary conditions”, SIAM J. Numer. Anal., 27 (1990), 608-621 | DOI | MR | Zbl

[19] Pani A.K., “An $H^1$-Galerkin mixed finite element method for parabolic partial differential equations”, SIAM J. Numer. Anal., 35:2 (1998), 712-727 | DOI | MR | Zbl

[20] Pani A.K., Fairweather G., “$H^1$-Galerkin mixed finite element method for parabolic partial integro-differential equations”, IMA J. Numer. Anal., 22 (2002), 231-252 | DOI | MR | Zbl

[21] Russell T.F., “Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media”, SIAM J. Numer. Anal., 22:5 (1985), 970-1013 | DOI | MR | Zbl

[22] Raviart P.A., Thomas J.M., “A mixed finite element method for second order elliptic problems”, Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., 606, Springer, Berlin, 1977, 292-315 | DOI | MR

[23] Shi F., Yu J.P., Li K.T., “A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair”, Int. J. Comput. Math., 88 (2011), 2293-2305 | DOI | MR | Zbl

[24] Sinha R.K., Ewing R.E., Lazarov R.D., “Mixed finite element approximations of parabolic integro-differential equations with nonsmooth initial data”, SIAM J. Numer. Anal., 47:5 (2009), 3269-3292 | DOI | MR | Zbl

[25] Thomée V., Zhang N.Y., “Error estimates for semidiscrete finite element methods for parabolic integro-differential equations”, Math. Comput., 53 (1989), 121-139 | DOI | MR | Zbl

[26] Wu L., Allen M.B., “A two-grid method for mixed finite-element solution of reaction-diffusion equations”, Numerical Methods for Partial Differential Equations, 15 (1999), 317-332 | 3.0.CO;2-U class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[27] Xu J., “A novel two-grid method for semilinear elliptic equations”, SIAM J. Sci. Comput., 15 (1994), 231-237 | DOI | MR | Zbl

[28] Xu J., “Two-grid discretization techniques for linear and nonlinear PDEs”, SIAM J. Numer. Anal., 33:5 (1996), 1759-1777 | DOI | MR | Zbl

[29] Xu J., Zhou A., “A two-grid discretization scheme for eigenvalue problems”, Math. Comput., 70 (2001), 17-25 | MR | Zbl