936 research outputs found
The DPG-star method
This article introduces the DPG-star (from now on, denoted DPG) finite
element method. It is a method that is in some sense dual to the discontinuous
Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to
solve an overdetermined discretization of a boundary value problem. In the same
vein, the DPG methodology is a means to solve an underdetermined
discretization. These two viewpoints are developed by embedding the same
operator equation into two different saddle-point problems. The analyses of the
two problems have many common elements. Comparison to other methods in the
literature round out the newly garnered perspective. Notably, DPG and DPG
methods can be seen as generalizations of and
least-squares methods, respectively. A priori error analysis and a posteriori
error control for the DPG method are considered in detail. Reports of
several numerical experiments are provided which demonstrate the essential
features of the new method. A notable difference between the results from the
DPG and DPG analyses is that the convergence rates of the former are
limited by the regularity of an extraneous Lagrange multiplier variable
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Computational Engineering
The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods
Variational formulation of the finite calculus equations in solid mechanics and diffusion-reaction problems
We present a variational formulation of the finite calculus (FIC) equations for problems in mechanics governed by differential equations with symmetric operators. Applications considered include solid mechanics, diffusion-transport and diffusion-reaction problems. The key of the variational formulation is the identification of the FIC governing equations with the classical differential equations of mechanics written in terms of modified non-local variables. A total potential energy (TPE) functional is found in terms of the modified variables. The FIC equations in the domain and the boundary are recovered as the Euler-Lagrange equations and the natural boundary condition of the TPE functional, respectively. Symmetric finite element equations are obtained after discretization of the TPE functional, therefore preserving the symmetry of the governing infinitesimal equations. The variational FIC expression is reinterpreted as a Petrov Galerkin weighted residual form of the original FIC equations with non-local weighting functions. The analogy of the variational FIC-FEM formulation with a discontinuous Galerkin method is recognized. Extensions to multidimensional linear elastostatics and diffusion-reaction problems are presented
Variational formulation of the finite calculus equations in solid mechanics and diffusion-reaction problems
We present a variational formulation of the finite calculus (FIC) equations for problems in mechanics governed by differential equations with symmetric operators. Applications considered include solid mechanics, diffusion-transport and diffusion-reaction problems. The key of the variational formulation is the identification of the FIC governing equations with the classical differential equations of mechanics written in terms of modified non-local variables. A total potential energy (TPE) functional is found in terms of the modified variables. The FIC equations in the domain and the boundary are recovered as the Euler-Lagrange equations and the natural boundary condition of the TPE functional, respectively. Symmetric finite element equations are obtained after discretization of the TPE functional, therefore preserving the symmetry of the governing infinitesimal equations. The variational FIC expression is reinterpreted as a Petrov Galerkin weighted residual form of the original FIC equations with non-local weighting functions. The analogy of the variational FIC-FEM formulation with a discontinuous Galerkin method is recognized. Extensions to multidimensional linear elastostatics and diffusion-reaction problems are presented.Preprin
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