Error analysis of the reproducing kernel particle method.

*(English)*Zbl 0992.65119The authors actually discuss a class of projective methods (generalizations of Bubnov-Galerkin methods) with special basis functions associated with the sets of points (quasi-grids) in the closure of the domain. This sets have the parameter \(r>0\). When it tends to zero, the sets (in the given examples) are special grids (only one-dimensional case is considered in details).

Specific properties of smooth basis functions are connected with the fact that several of them might have nonzero values at the same point. Such subspaces were discussed by J.-P. Aubin [Approximation of elliptic boundary-value problems (1972; Zbl 0248.65063)] devoted to approximation of elliptic problems. They were of help in constructing smooth approximations but it was clear that they meet with difficulties in case of the Dirichlet boundary conditions (the authors also underline this fact). Moreover, such approximations lead to systems of equations of rather involved type for which it is very difficult to suggest good iterative methods (some examples for biharmonic equation can be found in Chapter 8 of the reviewer’s book [Optimization in solving elliptic problems. CRC Press (1996; Zbl 0852.65087)].)

The authors present some results dealing with interpolation error estimates for special sets of points and basis functions. These quasi-grids have some regular properties (like standard grids) and allow to obtain estimates of the same type as in theory of spline approximations. The case of the Dirichlet boundary conditions is considered for ordinary differential equations of the second order. Numerical examples correspond to similar one-dimensional problems.

Specific properties of smooth basis functions are connected with the fact that several of them might have nonzero values at the same point. Such subspaces were discussed by J.-P. Aubin [Approximation of elliptic boundary-value problems (1972; Zbl 0248.65063)] devoted to approximation of elliptic problems. They were of help in constructing smooth approximations but it was clear that they meet with difficulties in case of the Dirichlet boundary conditions (the authors also underline this fact). Moreover, such approximations lead to systems of equations of rather involved type for which it is very difficult to suggest good iterative methods (some examples for biharmonic equation can be found in Chapter 8 of the reviewer’s book [Optimization in solving elliptic problems. CRC Press (1996; Zbl 0852.65087)].)

The authors present some results dealing with interpolation error estimates for special sets of points and basis functions. These quasi-grids have some regular properties (like standard grids) and allow to obtain estimates of the same type as in theory of spline approximations. The case of the Dirichlet boundary conditions is considered for ordinary differential equations of the second order. Numerical examples correspond to similar one-dimensional problems.

Reviewer: Evgenij D’yakonov (Moskva)

##### MSC:

65N15 | Error bounds for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35J40 | Boundary value problems for higher-order elliptic equations |

##### Keywords:

elliptic equations; projective methods; error analysis; reproducing kernel particle method; Bubnov-Galerkin methods; smooth basis functions; iteration methods; biharmonic equation; numerical examples
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\textit{W. Han} and \textit{X. Meng}, Comput. Methods Appl. Mech. Eng. 190, No. 46--47, 6157--6181 (2001; Zbl 0992.65119)

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