# zbMATH — the first resource for mathematics

Biquadratic finite volume element methods based on optimal stress points for parabolic problems. (English) Zbl 1242.65180
The authors consider the following boundary value problem: $\frac{\partial u}{\partial t} - \nabla(a(x)\nabla u) = f(x,t),\;\;(x,t) \in \Omega \times (0, T],$
$u(x,t)=0, (x,t)\in \partial \Omega \times (0,T], \;\;u(x, 0)=u_0(x), \;\;x \in \Omega$ where $$\Omega$$ is a rectangle, $$x = (x,y)$$, $$a(x)$$ is a positive real-valued function and $$f(x,t):\Omega \rightarrow \mathbb{R}$$. The initial function $$u_0(x)$$, the functions $$a(x), f(x,t)$$ are assumed to be sufficiently smooth to ensure that the above problem has a unique solution in the appropriate Sobolev space. Semi-discrete and discrete finite volume element (FVE) schemes based on optimal stress points are constructed. Optimal order error estimates in $$H^1$$ and $$L^2$$ norms are derived. Also obtained are the superconvergence of numerical gradients at optimal stress points. The theoretical results are confirmed through a numerical experiment for $$a(x)=exp(x+y)$$ when the exact solution of the problem is $$u(x,f)=e^{-t}\sin(x)\sin(y)$$.

##### MSC:
 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text:
##### References:
 [1] Cai, Z.Q., On the finite volume element method, Numer. math., 58, 7, 713-735, (1991) · Zbl 0731.65093 [2] Cai, Z.Q.; Mandel, J.; McCormick, S., The finite volume element method for diffusion equations on general triangulations, SIAM J. numer. anal., 28, 2, 392-402, (1991) · Zbl 0729.65086 [3] Ewing, R.; Lazarov, R.E.; Lin, Y.P., Finite volume element approximations of nonlocal reactive flows in porous media, Numer. methods partial differential equations, 16, 3, 285-311, (2000) · Zbl 0961.76050 [4] Huang, J.G.; Xi, S.T., On the finite volume element method for general self-adjoint elliptic problems, SIAM J. numer. anal., 35, 5, 1762-1774, (1998) · Zbl 0913.65097 [5] Li, R.H.; Chen, Z.Y.; Wu, W., Generalized difference methods for differential equations, (2000), Marcel Dekker New York [6] Li, Y.H.; Li, R.H., Generalized difference methods on arbitrary quadrilateral networks, J. comput. math., 17, 6, 653-672, (1999) · Zbl 0946.65098 [7] Wu, W.; Li, R.H., A generalized difference method for solving one-dimensional second-order elliptic and parabolic differential equations, Chinese ann. math. ser. A, 5, 3, 303-312, (1984), (in Chinese) · Zbl 0538.65073 [8] Chou, S.H.; Li, Q., Error estimates in $$L^2, H^1$$ and $$L^\infty$$ in covolume methods for elliptic and parabolic problems: a unified approach, Math. comp., 69, 229, 103-120, (2000) · Zbl 0936.65127 [9] Chou, S.H.; Kwak, D.Y.; Li, Q., $$L^p$$ error estimates and superconvergence for covolume or finite volume element methods, Numer. methods partial differential equations, 19, 4, 463-486, (2003) · Zbl 1029.65123 [10] Bank, R.E.; Rose, D.J., Some error estimates for the box method, SIAM J. numer. anal., 24, 4, 777-787, (1987) · Zbl 0634.65105 [11] Xu, J.C.; Zou, Q.S., Analysis of linear and quadratic simplicial finite volume methods for elliptic equations, Numer. math., 111, 3, 469-492, (2009) · Zbl 1169.65110 [12] Chen, Z.Y.; Li, R.H.; Zhou, A.H., A note on the optimal $$L^2$$-estimate of the finite volume element method, Adv. comput. math., 16, 4, 291-303, (2002) · Zbl 0997.65122 [13] Lv, J.L.; Li, Y.H., $$L^2$$ error estimate of the finite volume element methods on quadrilateral meshes, Adv. comput. math., 33, 2, 129-148, (2010) · Zbl 1198.65218 [14] Yang, M.; Chen, C., ADI quadratic finite volume element methods for second order hyperbolic problems, J. appl. math. comput., 31, 1-2, 395-411, (2009) · Zbl 1179.65128 [15] Plexousakis, M.; Zouraris, G.E., On the construction and analysis of high order locally conservative finite volume-type methods for one-dimensional elliptic problems, SIAM J. numer. anal., 42, 3, 1226-1260, (2004) · Zbl 1083.65074 [16] Yu, C.H.; Li, Y.H., Biquadratic element finite volume method based on optimal stress points for solving Poisson equations, Math. numer. sin., 32, 1, 59-74, (2010), (in Chinese) · Zbl 1224.65244 [17] Wang, T.K.; Gu, Y.S., Superconvergent biquadratic finite volume element method for two-dimensional poisson’s equations, J. comput. appl. math., 234, 2, 447-460, (2010) · Zbl 1191.65143 [18] Gao, G.H.; Wang, T.K., Cubic superconvergent finite volume element method for one-dimensional elliptic and parabolic equations, J. comput. appl. math., 233, 9, 2285-2301, (2010) · Zbl 1197.65171 [19] Ma, X.L.; Shu, S.; Zhou, A.H., Symmetric finite volume discretizations for parabolic problems, Comput. methods appl. mech. engrg., 192, 39-40, 4467-4485, (2003) · Zbl 1038.65079 [20] Rui, H.X., Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems, J. comput. appl. math., 146, 2, 373-386, (2002) · Zbl 1020.65066 [21] Chatzipantelidis, P.; Lazarov, R.D.; Thomée, V., Error estimates for a finite volume element method for parabolic equations in convex polygonal domains, Numer. methods partial differential equations, 20, 5, 650-674, (2004) · Zbl 1067.65092 [22] Chatzipantelidis, P.; Lazarov, R.D.; Thomée, V., Parabolic finite volume element equations in nonconvex polygonal domains, Numer. methods partial differential equations, 25, 3, 507-525, (2009) · Zbl 1168.65051 [23] Bradji, A., Some simple error estimates of finite volume approximate solution for parabolic equations, C. R. math. acad. sci. Paris, 346, 9-10, 571-574, (2008) · Zbl 1142.65075 [24] Eymard, R.; Fuhrmann, J.; Gärtner, K., A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems, Numer. math., 102, 3, 463-495, (2006) · Zbl 1116.65101 [25] Chen, Z.Y., A generalized difference method for equations of heat conduction, Acta sci. natur. univ. sunyatseni, 29, 1, 6-13, (1990), (in Chinese) [26] Wang, T.K., High accuracy finite volume element methods for one-dimensional second-order elliptic and parabolic differential equations, J. numer. methods comput. appl., 23, 4, 264-274, (2002) [27] Yang, M.; Yuan, Y., A symmetric characteristic FVE method with second order accuracy for nonlinear convection diffusion problems, J. comput. appl. math., 200, 2, 677-700, (2007) · Zbl 1121.65103 [28] Chen, C.M., Structure theory of superconvergence of finite elements, (2001), Hunan Press of Science and Technology Changsha, (in Chinese) [29] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.