We discuss stabilized Galerkin approximations in a new framework, widening the scope from the usual dichotomy of the discontinuous Galerkin method on the one hand and Petrov– Galerkin methods such as the SUPG method on the other. The idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation, thus circumventing the n...

In this paper, we first introduce fractional integral spaces, which possess some features: (i) when 0 < α < 1, functions in these spaces are not required to be zero on the boundary; (ii)the tempered fractional operators are equivalent to the Riemann-Liouville operator in the sense of the norm. Spectral Galerkin and Petrov-Galerkin methods for tempered fractional advection problems and tempered ...

These lectures present a relatively recent introduction into the class of discontinuos Galerkin (DG) methods, named Discontinuous Petrov-Galerkin (DPG) methods. DPG methods, in which DG spaces form a critical ingredient, can be thought of as least-square methods in nonstandard norms, or as Petrov-Galerkin methods with special test spaces, or as a nonstandard mixed method. We will pursue all the...

Motivated by the Discontinous Petrov-Galerkin method from [Numer. Methods Partial Differential Equations, 27 (2011), 70–105] by Demkowicz and Gopalakrishnan, we study a variational formulation of second order elliptic equations in mixed form, that is obtained by piecewise integrating one of the two equations in the system w.r.t. a partition of the domain into mesh cells. We apply a Petrov-Galer...

In this work, we revisit the spectral Petrov-Galerkin method for fractional elliptic equations with general operators. To prove optimal convergence of method, first present ultra-weak formulation and establish its well-posedness. Then, based on such a novel formulation, are able to discrete counterpart obtain in weighted L2-norm. For simple easy implementation also describe fast solver linear s...

In this article we derive local error estimates for some Petrov-Galerkin methods applied to strongly elliptic equations on smooth curves of the plane. The results, e.g., cover the basic first-kind and second-kind integral equations appearing in the boundary element solution of the potential problem. The discretization model includes the Galerkin method and the collocation method using smoothest...

A meshless local Petrov-Galerkin (MLPG) method is applied to solve problems of Reissner-Mindlin shells under thermal loading. Both stationary and thermal shock loads are analyzed here. Functionally graded materials with a continuous variation of properties in the shell thickness direction are considered here. A weak formulation for the set of governing equations in the Reissner-Mindlin theory i...