The discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [15, 17] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. A key question that has not yet been answered in general—though there are some results for Poisson, e.g.—is how best to precondition the DPG system matrix, so that iterative solvers ...

We consider variational inequalities with different trial and test spaces and a possibly non-coercive bilinear form. Well-posedness is shown under general conditions that are e.g. valid for the space-time variational formulation of parabolic variational inequalities. Moreover, we prove an estimate for the error of a Petrov-Galerkin approximation in terms of the residual. For parabolic variation...

We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant βδ: βδ is bounded from below by unity for the heat equation; βδ grows only linearly in time for non-coercive (asymptotically stable) convection operators. The latter in turn p...

A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linear ordinary differential equation. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by the Gauss quadrature rule in the formulation itself. Optimal order a priori error estimates are obtai...

Isogeometric Analysis (IGA), in combination with the Streamline Upwind Petrov– Galerkin (SUPG) stabilization, is studied for the discretization of steady-state convection-diffusion equations. Numerical results obtained for the Hemker problem are compared with results computed with the SUPG finite element method of the same order. Using an appropriate parameterization for IGA, the computed solut...

In this paper, we introduce the Petrov-Galerkin method for solution of stochastic Volterra integral equations. Here, we use continues Lagrange-type k-0 elements, since these elements have simple structure and via them, the solution of stochastic Volterra integral equation is reduced to algebraic equations. Also the error analysis of this method is done. In Comparison with other methods, this me...

We propose and analyse a fully discrete Petrov–Galerkin method with quadrature, for solving second-order, variable coefficient, elliptic boundary value problems on rectangular domains. In our scheme, the trial space consists of C2 splines of degree r 3, the test space consists of C0 splines of degree r − 2, and we use composite (r − 1)-point Gauss quadrature. We show existence and uniqueness of...

The aim of this paper is twofold. First, we review the recent Petrov-Galerkin enriched method (PGEM) to stabilize numerical solutions of BVP’s in primal and mixed forms. Then, we extend such enrichment technique to a mixed singularly perturbed problem, namely, the generalized Stokes problem, and focus on a stabilized finite element method arising in a natural way after performing static condens...

in this paper, three dimensional (3d) static and dynamic analysis of thick plates based on the meshless local petrov-galerkin (mlpg) is presented. using the kinematics of a three-dimensional continuum, the local weak form of the equilibrium equations is derived. a weak formulation for the set of governing equations is transformed into local integral equations on local sub-domains by using a uni...

Modifications to the standard finite element mass matrix are considered with the aim of preserving the positivity of the discrete solution. The approach is used in connection with calculating the initial time derivative values for parabolic equations and in connection with nonlinear Petrov-Galerkin schemes for hyperbolic equations in one space dimension. The extension of the ideas to unstructur...