Two-Grid Finite Volume Element Method for Linear and Nonlinear Elliptic Problems. Bi, C. & Ginting, V. Numerische Mathematik, 108(2):177-198, 2007.
Paper doi abstract bibtex Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates.
@article {MR2358002,
AUTHOR = {Bi, C. and Ginting, V.},
TITLE = {Two-{G}rid {F}inite {V}olume {E}lement {M}ethod for {L}inear and {N}onlinear
{E}lliptic {P}roblems},
JOURNAL = {Numerische Mathematik},
VOLUME = {108},
YEAR = {2007},
NUMBER = {2},
PAGES = {177-198},
ISSN = {0029-599X},
CODEN = {NUMMA7},
MRCLASS = {65N06},
MRNUMBER = {2358002 (2008i:65227)},
MRREVIEWER = {Zheng Hui Xie},
DOI = {10.1007/s00211-007-0115-9},
URL = {http://dx.doi.org/10.1007/s00211-007-0115-9},
ABSTRACT="Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates."
}
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