Author
V. A. Rukavishnikov
Bio: V. A. Rukavishnikov is an academic researcher from Russian Academy of Sciences. The author has contributed to research in topics: Singularity & Finite element method. The author has an hindex of 9, co-authored 45 publications receiving 282 citations.
Papers published on a yearly basis
Papers
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TL;DR: The existence and uniqueness of the R"@n-generalized solution for the third-boundary-value problem and the non-self-adjoint second-order elliptic equation with strong singularity are established and a finite element method with a basis containing singular functions is constructed.
39 citations
TL;DR: The weighted analogue of the Ladyzhenskaya–Babuska–Brezzi condition is proved and a new weighted finite element method is constructed that shows the efficiency of the method.
28 citations
25 citations
25 citations
21 citations
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Journal Article•
1,019 citations
01 Jan 2001
TL;DR: This work constructs a simple and efficient adaptive FEM for elliptic partial differential equations and proves that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants.
Abstract: Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.
Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, Stokes, Uzawa
AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20
Published: SIAM Review, 44 (2002) 631--658.
337 citations
TL;DR: In this paper, a mathematical model of the problem of nonlinear oscillations of a viscoelastic pipeline conveying fluid is developed in the Boltzmann-Volterra integral model with weakly singular kernels of heredity.
46 citations
TL;DR: The existence and uniqueness of the R"@n-generalized solution for the third-boundary-value problem and the non-self-adjoint second-order elliptic equation with strong singularity are established and a finite element method with a basis containing singular functions is constructed.
39 citations
TL;DR: A notion of R"@n-generalized solution to time-harmonic Maxwell equations with strong singularity in a 2D nonconvex polygonal domain is introduced and a new weighted edge FEM is developed.
39 citations