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Adaptation of the finite element method for mathematical model with nonsmooth solutions. (Russian. English summary) Zbl 1356.65201

Summary: The method of finite elements adapted to the boundary value problem of fourth order derivative, which arises in modelling small deformations of the rod with localized features (elastic supports, impulse of external influence) that is located along the segment \([0;1]\). Such features lead to the loss of smoothness of solutions. To overcome the difficulties that arise in this case, we use pointwise method of interpreting the differential equation proposed by Yu. V. Pokornyj [Dokl. Math. 59, No. 1, 34–37 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 364, No. 2, 167–169 (1999; Zbl 0962.34006); Dokl. Math. 65, No. 2, 262–265 (2002); translation from Dokl. Akad. Nauk 383, No. 5, 601–604 (2002; Zbl 1197.34168)]. This method is proved to be effective for the second order boundary value problems with non-smooth and discontinuous solutions.

MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74S05 Finite element methods applied to problems in solid mechanics
65L70 Error bounds for numerical methods for ordinary differential equations
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