Three different truly Meshless Local Petrov-Galerkin (MLPG) methods are developed for solving 3D elasto-static problems. Using the general MLPG concept, these methods are derived through the local weak forms of the equilibrium equations, by using different test functions, namely, the Heaviside function, the Dirac delta function, and the fundamental solutions. The one with the use of the fundame...

We introduce a space-time discretization for linear first-order hyperbolic evolution systems using a discontinuous Galerkin approximation in space and a Petrov–Galerkin scheme in time. We show well-posedness and convergence of the discrete system. Then we introduce an adaptive strategy based on goal-oriented dual-weighted error estimation. The full space-time linear system is solved with a para...

In this paper, we combine continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time stepping schemes with local projection method applied to inf-sup stable discretization of the transient Oseen problem. Using variational-type time-discretization methods of polynomial degree k, we show that the cGP(k) and dG(k) methods are accurate of order k+1, in the whole time interval. Moreover, ...

We lay out a program for constructing discontinuous Petrov-Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well-posedness of the undiscretized problem. Although the question of stable test space choice had attracted the atten...

In Meshless natura1 neighbour Petrov-Galerkin method, The natural neighbour interpolation is used as trial function and a weak form over the local polygonal sub-domains constructed by Delaunay triangular is used to obtain the discretized system of equilibrium equations, and it’s a new truly meshless method. This method simplified the formation of the equilibrium equations, facilitates the impos...

A new symmetric local projection method built on residual bases (RELP) makes linear equal-order finite element pairs stable for the Darcy problem. The derivation is performed inside a Petrov-Galerkin enriching space approach (PGEM) which indicates parameter-free terms to be added to the Galerkin method without compromising consistency. Velocity and pressure spaces are augmented using solutions ...

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...

Novel non-hyper-singular [i.e., only strongly-singular] boundary-integral-equations for the gradients of the acoustic velocity potential, involving only O(r−2) singularities at the surface of a 3-D body, are derived, for solving problems of acoustics governed by the Helmholtz differential equation. The gradients of the fundamental solution to the Helmholtz differential equation for the velocity...

We give a convergence estimate for a Petrov-Galerkin Algebraic Multigrid method. In this method, the prolongations are defined using the concept of smoothed aggregation while the restrictions are simple aggregation operators. The analysis is carried out by showing that these methods can be interpreted as variational Ritz-Galerkin ones using modified transfer and smoothing operators. The estimat...