Treatment of material discontinuity in two meshless local Petrov-Galerkin (MLPG) formulations of axisymmetric transient heat conduction.

*(English)*Zbl 1075.80001The authors compare the efficiency of two Meshless Local Petrov-Galerkin numerical methods (MLPG) in order to solve some heat equation posed in two concentric disks made of two different materials. Writing the two heat equations in polar coordinates, the authors impose the continuity of the temperatures and that of the fluxes accross the interface between the two disks. These two temperatures start from initial data which satisfy similar continuity conditions. A given temperature is imposed on the outer boundary. The numerical resolution of this problem is built on the construction of trial functions chosen as: \(T_{h}\left( r,t\right) =\sum_{j=1}^{m}r^{j-1}a_{j}\left( r,t\right) \) where the coefficients \(a_{j}\) are obtained when minimizing some functional \(J\) which involves a weight taken as a fourth order spline function based on the nodes \(r_{i}\) and on some functions \(\widehat{T}_{i}\left( t\right) \) which are fictitious values of the temperature at \(r=r_{i}\). The main part of the paper presents some numerical computations obtained using two MLPG methods which are considered (MLPG1 and MLPG5). The conclusion is that the two methods are quite efficient in order to handle the discontinuity of the gradient at the interface between the two disks. A small advantage is observed for the MLPG5 for the dependance of the \(H^{0}\) or of the \(H^{1}\) norm of the error with respect to the number of nodes \(r_{i}\). A table compares the MLPG, the FEM and the EFG methods for the numerical resolution of such problems.

Reviewer: Alain Brillard (Mulhouse)

##### MSC:

80A20 | Heat and mass transfer, heat flow (MSC2010) |

80M25 | Other numerical methods (thermodynamics) (MSC2010) |

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

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\textit{R. C. Batra} et al., Int. J. Numer. Methods Eng. 61, No. 14, 2461--2479 (2004; Zbl 1075.80001)

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