This paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamlineupwind Petrov–Galerkin (SUPG) and local projection stabilization (LPS) meth-ods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) meth-ods and continuo...

A level set method combining features of front tracking methods and fixed domain methods is presented to model microstructure evolution in the solidification of multi-component multiphase alloy systems. Phase boundaries are explicitly tracked by solving the multi-phase level set equations. Diffused interfaces are constructed by extending a small width in both directions from these explicitly tr...

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...

We discuss the (conjugated) Bubnov–Galerkin and Petrov–Galerkin infinite element (IE) discretizations to Helmholtz equation including the use of elements of locally variable order, optimal choice of IE shape functions, calculation of Echo Area (EA), and automatic hp-adaptivity. The discussion is illustrated with 2D numerical experiments. 2005 Elsevier B.V. All rights reserved. MSC classificatio...

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace – and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal...

Using the directly derived non-hyper singular integral equations for displacement gradients [as in Okada, Rajiyah, and Atluri (1989a)], simple and straightforward derivations of weakly singular traction BIE’s for solids undergoing small deformations are presented. A large number of “intrinsic properties” of the fundamental solutions in elasticity are developed, and are used in rendering the tBI...

It is known that the standard Galerkin finite element method (SGFEM) based on low order piecewise polynomials is unsuitable for the solution of steady and unsteady advection diffusion problems. In case the advection term dominates the diffusion one or small time steps are employed, numerical solutions obtained by SGFEM suffer from nonphysical oscillations, unless appropriately designed meshes a...

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace—and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Ga...

We present an efficient implementation of the double adaptivity algorithm Cohen et al. (2012) within setting Petrov–Galerkin method with optimal test functions. apply this to ultraweak variational formulation a general linear problem discretized standard Galerkin finite element method. As example, we demonstrate feasibility in context convection-dominated diffusion problem. The presented ideas,...