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Higher-order compact mixed methods. (English) Zbl 0885.65109
The authors construct a class of mixed higher-order finite difference schemes on compact stencils for second-order elliptic partial differential equations. Dirichlet and Neumann boundary conditions for the 2D problem are considered. The resulting difference solutions are \(O(h^4)\) accurate in both \(u\) and the flux \(\sigma= \nabla u\) at the grid points, where \(u\) is the solution of the corresponding boundary value problem. The results of some numerical experiments are presented to demonstrate the superconvergence rates.
Reviewer: V.Makarov (Kyïv)

65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI
[1] Arbogast, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comp. 18 (1997)
[2] Brezzi, Mixed and Nonconforming Finite Element Methods (1991) · Zbl 0788.73002
[3] Ewing, Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal. 28 pp 1015– (1991) · Zbl 0733.65065
[4] Hirsh, Higher order accurate difference solution of fluid mechanics problems by a compact differencing technique, J. Comput. Phys. 9 (1) pp 90– (1975) · Zbl 0326.76024
[5] MacKinnon, Analysis of material interface discontinuities and superconvergent fluxes in finite difference theory, J. Comput. Phys. 75 (1) pp 151– (1988) · Zbl 0632.76110
[6] MacKinnon, Superconvergent derivatives: A Taylor series analysis, Int. j. numer. methods eng. 28 pp 489– (1989) · Zbl 0709.73073
[7] MacKinnon, Nodal superconvergence and solution enhancement for a class of finite element and finite difference methods, SIAM J. Sci. Stat. Comput. 11 (2) pp 343– (1990) · Zbl 0701.65071
[8] Spotz, High-order compact finite difference schemes for computational mechanics (1995)
[9] Spotz, High-order compact scheme for the streamfunction vorticity equations, Int. j. numer. methods eng. 38 (20) pp 3497– (1995) · Zbl 0836.76065
[10] Spotz, A high-order compact formulation for the 3D Poisson equation, Numer. Methods Partial Diff. Equ. 12 pp 235– (1996) · Zbl 0866.65066
[11] Pereyra, Highly accurate numerical solution of quasilinear elliptic boundary-value problems in n dimensions, Math. Comput. 24 pp 771– (1970) · Zbl 0219.65084
[12] Adams, Recent enhancements in MUDPACK, a multigrid software package for elliptic partial differential equations, Appl. Math. Comput. 43 (1) pp 79– (1991) · Zbl 0727.65104
[13] Hemker, Lecture Notes in Mathematics, in: Multigrid Methods: Proceedings, Küln-Porz, 1981 pp 485– (1982)
[14] Hemker, Multigrid Methods II: Proceedings of 2nd European Conference on Multigrid Methods pp 149– (1985)
[15] Stetter, The defect correction principle and discretization methods, Numer. Math. 29 pp 425– (1978) · Zbl 0362.65052
[16] Carey, Computational Grids: Generation, Adaption and Solution Strategies (1997) · Zbl 0955.74001
[17] Axelsson, On the numerical solution of two-point singularly perturbed boundary-value problems, Comput. Methods Appl. Mech. Eng. 50 pp 217– (1985) · Zbl 0553.76034
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