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Application of the Monte Carlo method with meshless random walk procedure to selected scalar elliptic problems

Milewski S.

Archives of Mechanics

2019

Vol. 71, nr 4-5

337–-375

The combined stochastic-deterministic approach, which may be applied to the numerical analysis of a wide range of scalar elliptic problems of civil engineering, is presented in this paper. It is based on the well-known Monte Carlo concept with a random walk procedure, in which series of random paths are constructed. Additionally, it incorporates selected features of the meshless finite difference method, especially star selection criteria and a local weighted function approximation. The approach leads to the explicit stochastic formula relating one unknown function value with all a-priori known data parameters. Therefore, it allows for a fast and effective estimation of the solution value at the selected point(s), without the necessity of generation of large systems of equations, combining all unknown values. In such a manner, the proposed approach develops and extends the original standard Monte Carlo one toward analysis of boundary value problems with more complex shape geometry, natural boundary conditions, non-homogeneous right-hand sides as well as anisotropic and non-linear material models. The paper is illustrated with numerical results of selected elliptic problems, including a torsion problem of a prismatic bar, a stationary heat flow analysis with anisotropic and non-linear material functions, as well as an inverse heat problem. Moreover, the appropriate coupling with other deterministic methods (e.g., the finite element method) is considered.

Fictitious domain approach for numerical solution of elliptic problems

Myśliński A., Żochowski A.

Control and Cybernetics

2000

Vol. 29, no 1

305-323

In the paper the numerical aspects of the fictitious domain method for elliptic problems are considered. Theoretical results concerning the equivalence of original and embedded domain elliptic problems as well as the convergence of discretization are recalled. The dependence of accuracy of numerical solutions to elliptic problems on the approximation of boundary conditions as well as on the order of shape functions are disscussed. The numerical examples are provided.