In order to improve the approximation of spatial derivatives without meshes, a set of meshfree numerical schemes for derivative terms is developed, which is compatible with the coordinates of Cartesian, cylindrical, and spherical. Based on the comparisons between numerical and theoretical solutions, errors and convergences are assessed by a posteriori method, which shows that the approximations...

The a posteriori error estimation of finite element approximations of elliptic boundary value problems has reached some state of maturity, as it is documented by a variety of monographs on this subject (cf., e.g., [1, 2, 3, 4, 5]). There are different concepts such as • residual type a posteriori error estimators, • hierarchical type a posteriori error estimators, • error estimators based on lo...

The mixtures of experts (ME) model ooers a modular structure suitable for a divide-and-conquer approach to pattern recognition. It has a probabilistic interpretation in terms of a mixture model, which forms the basis for the error function associated with MEs. In this paper, it is shown that for classiication problems the minimization of this ME error function leads to ME outputs estimating the...

This paper considers the model problem in two-dimensional domain. The a posteriori error indicator for p-version Finite Element Methods (FEM) is discussed, and the reliable property of this a posteriori error indicator is investigated. Specially, we reformulate the a posteriori error indicator by orthogonal polynomials, which is easily used in practical applications.