Efficient cubature formulae for MLPG and related methods

The paper introduces four kinds of compact, simple to implement Gaussian cubature formulae for approximating the domain integrals arising in the discrete local weak form (DLWF) of a governing partial differential equation solved by means of the meshless local Petrov–Galerkin method of type MLPG1. The integral weight functions are fixed to be the quartic-spline weight function of the moving least squares (MLS) method and the function’s gradient. The integration domain is a circle in 2D or a sphere in 3D. The fact that the DLWF test functions are directly incorporated into the formulae increases both their exactness degree and their computational efficiency. A number of numerical tests are carried out in order to asses the accuracy of the cubature formulae. For integrands involving MLS shape functions, the main factor controlling the integration accuracy is found to be the accuracy of the MLS-approximation. Only a small number of cubature points is thus required to match that accuracy without a need for domain partitioning. The recommended approach for increasing the overall accuracy is by adding more MLS nodes and taking advantage of the computationally inexpensive cubature formulae