A symmetric mixed covolume method for the nonlinear parabolic problem

In this paper, we develop a symmetric mixed covolume scheme for the nonlinear parabolic equations. The existence and uniqueness of the scheme are proved by using the lowest order Raviart–Thomas mixed element space on rectangular grids. We carry out a rigorous error estimates for the constructed scheme, establishing the first-order convergence for both velocity and pressure. Numerical examples are provided to verify that the numerical results are all in consistent with the theoretical analysis.

[1]  M. Safari,et al.  Adaptive control design for a nonlinear parabolic PDE: Application to water coning , 2018 .

[2]  H. Rui,et al.  An approximation of incompressible miscible displacement in porous media by mixed finite elements and symmetric finite volume element method of characteristics , 2013 .

[3]  H. Rui.  Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems , 2002 .

[4]  Shuhuang Xiang,et al.  A Collocation Boundary Value Method for Linear Volterra Integral Equations , 2017, J. Sci. Comput..

[5]  Eun-Jae Park,et al.  Mixed finite element methods for nonlinear second-order elliptic problems , 1995 .

[6]  Ziwen Jiang,et al.  Mixed covolume method for parabolic problems on triangular grids , 2009, Appl. Math. Comput..

[7]  Jiming Wu,et al.  Vertex-Centered Linearity-Preserving Schemes for Nonlinear Parabolic Problems on Polygonal Grids , 2017, J. Sci. Comput..

[8]  Hong Li,et al.  A Splitting Mixed Covolume Method for Viscoelastic Wave Equations on Triangular Grids , 2020, Mediterranean Journal of Mathematics.

[9]  Huan Liu,et al.  A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection-diffusion equations , 2020, Comput. Math. Appl..

[10]  Wanfu Tian,et al.  Superconvergence of mixed covolume method on quadrilateral grids for elliptic problems , 2012, J. Syst. Sci. Complex..

[11]  Chunjia Bi.  Superconvergence of mixed covolume method for elliptic problems on triangular grids , 2008 .

[12]  Panagiotis Chatzipantelidis,et al.  A Finite Volume Element Method for a Nonlinear Parabolic Problem , 2013 .

[13]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[14]  Catherine Choquet,et al.  Optimal Control for a Groundwater Pollution Ruled by a Convection–Diffusion–Reaction Problem , 2017, J. Optim. Theory Appl..

[15]  L. D. Marini.  An Inexpensive Method for the Evaluation of the Solution of the Lowest Order Raviart–Thomas Mixed Method , 1985 .

[16]  Numerical modeling of hyperbolic dominant transient fluid flow in saturated fractured rocks using Darcian approach , 2018, Groundwater for Sustainable Development.

[17]  Hong Li,et al.  Numerical Solution of Burgers’ Equation Based on Mixed Finite Volume Element Methods , 2020 .

[18]  Xinlong Feng,et al.  A stabilized extremum‐preserving scheme for nonlinear parabolic equation on polygonal meshes , 2019, International Journal for Numerical Methods in Fluids.

[19]  H. Rui.  Symmetric mixed covolume methods for parabolic problems , 2002 .

[20]  Yuanyuan Zhang,et al.  A two-grid finite element method for nonlinear parabolic integro-differential equations , 2018, Int. J. Comput. Math..

[21]  Hongxing Rui.  Superconvergence of a Mixed Covolume Method for Elliptic Problems , 2003, Computing.

[22]  Kwang Y. Kim,et al.  Mixed Covolume Methods for Quasi-Linear Second-Order Elliptic Problems , 2000, SIAM J. Numer. Anal..

[23]  Eun-Jae Park,et al.  Mixed finite element domain decomposition for nonlinear parabolic problems , 2000 .

[24]  Do Y. Kwak,et al.  Mixed Covolume Methods on Rectangular Grids For Elliptic Problems , 2000, SIAM J. Numer. Anal..

[25]  Chuanjun Chen,et al.  Two-grid methods for finite volume element approximations of nonlinear parabolic equations , 2009 .

[26]  Qing Yang,et al.  A discontinuous mixed covolume method for elliptic problems , 2011, J. Comput. Appl. Math..