Abstract. We present a Discontinuous Petrov-Galerkin method (DPG) for finite element discretization scheme of second order elliptic boundary value problems. The novel approach emanates from a one-element weak formulation of the differential problem (that is typical of Discontinuous Galerkin methods (DG)) which is based on introducing variables defined in the interior and on the boundary of the ...

Numerical solutions obtained by the Meshless Local Petrov-Galerkin (MLPG) method are presented for two dimensional steady-state heat conduction problems. The MLPG method is a truly meshless approach, and neither the nodal connectivity nor the background mesh is required for solving the initial-boundary-value problem. The penalty method is adopted to efficiently enforce the essential boundary co...

We develop a family of Eulerian-Lagrangian localized adjoint methods for the solution of the initial-boundary value problems for rst-order advection-reaction equations on general multi-dimensional domains. Diierent tracking algorithms, including the Euler and Runge-Kutta algorithms, are used. The derived schemes naturally incorporate innow boundary conditions into their formulations and do not ...

We discuss numerical properties of continuous Galerkin-Petrov and discontinuous Galerkin time discretizations applied to the heat equation as a prototypical example for scalar parabolic partial differential equations. For the space discretization, we use biquadratic quadrilateral finite elements on general two-dimensional meshes. We discuss implementation aspects of the time discretization as w...

A spacetime discontinuous Petrov-Galerkin (DPG) method for the linear wave equation is presented. This method is based on a weak formulation that uses a broken graph space. The wellposedness of this formulation is established using a previously presented abstract framework. One of the main tasks in the verification of the conditions of this framework is proving a density result. This is done in...

For the unsteady convection-diiusion equation in two dimensions we derive a new cell-based semi-discretization which is founded on the method of lines and a nite volume approach. Moreover, we present a second semidiscretization technique, a nonconforming Petrov-Galerkin method with exponentially tted trial and test functions. If we use appropriate quadrature rules both approaches are equivalent...

A novel spectral element technique is examined in which the test functions are different from the approximating elements. Examples are given for a simple 1D Helmholtz equation, Burger's Equation with a small viscosity, and for Darcy's Equation with a discontinous hydraulic conductivity. 1. Polynomial Approximation. Spectral element approximations have been shown to be an eeective tool for the a...

The main focus of this paper is on an a-posteriori analysis for different model-order strategies applied to optimal control problems governed by linear parabolic partial differential equations. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the reduced-order model, is from the (unknown) exact one. For the model-order reduction, H2,α-norm op...

This paper introduces a novel lowest-order discontinuous Petrov Galerkin (dPG) finite element method (FEM) for the Poisson model problem. The ultra-weak formulation allows for piecewise constant and affine ansatz functions and for piecewise affine and lowest-order Raviart-Thomas test functions. This lowest-order discretization for the Poisson model problem allows for a direct proof of the discr...

We present the stability and error analysis of the unified Petrov-Galerkin spectral method, developed in [1], for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any (1 + d)-dimensional space-time hypercube, d = 1, 2, 3, · · · , subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence and uniquene...