A short introduction is given about direct variational methods and their relation to Galerkin and moment methods, all flexible and powerful approaches for finding approximate solutions to difficult physical equations. An application of these methods is given in the form of the variational problem of minimizing the discomfort experienced during different journeys, between two fixed horizontal po...

Consider a model eigenvalue problem with a piecewise constant coefficient. We split the domain at the discontinuity of the coefficient function and define the multidomain variational formulation for the eigenproblem. The discrete multidomain variational formulations are defined for Legendre–Galerkin and Legendre-collocation methods. The spectral rate of convergence of the approximate eigensolut...

This paper deals with intrusive Galerkin projection methods with Roe-type solver for uncertain hyperbolic systems using a finite volume discretization in physical space and a piecewise continuous representation at the stochastic level. The aim of this paper is to design a cost-effective adaptation of the deterministic Dubois and Mehlman corrector to avoid entropy-violating shocks in the presenc...

We propose and analyze a symmetric weighted interior penalty (SWIP) method to approximate in a Discontinuous Galerkin framework advection-diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high Péclet numbers) o...

Coercivity is an important concept for proving existence and uniqueness of solutions to variational problems in Hilbert spaces. But, while the existence of coercivity estimates is well known for many variational problems arising from partial differential equations, it is still an open problem in the context of boundary integral operators arising from acoustic scattering problems, where rigorous...

We consider weakly singular integral equations of the rst kind on screens in IR 3. To obtain approximate solutions we use the hand p-versions of the Galerkin Boundary Element Method. We introduce two-level additive Schwarz operators with bounded condition numbers. Based on these operators we derive an a posteriori error estimate for the diierence between the exact solution and the approximate s...

A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions linear ordinary differential equations, in particular D-finite functions. It relies on an a posteriori scheme, where such bound computed afterwards, independently from how the approximation was built. Contrary Newton–Galerkin methods, widely used mathemati...

The collocation method and Galerkin method using parabolic splines are considered. Special adaptive meshes whose number of knots is independent of the small parameter of the problem are used. Unimprovable estimates in the L∞-norm are obtained. For the Galerkin method these estimates are quasioptimal, while for the collocation method they are suboptimal. Introduction It is well known that the sp...

This paper describes a modal analysis technique to approximate the vibrations of incompressible elastic solids using stabilized finite element method associated eigenvalue problem. It is explained why residual based formulations are not appropriate in this case, and formulation involving only pressure gradient employed. The effect stabilization term compared Galerkin approach detailed, both der...

Abstract. We introduce generalized Galerkin variational integrators, which are a natural generalization of discrete variational mechanics, whereby the discrete action, as opposed to the discrete Lagrangian, is the fundamental object. This is achieved by approximating the action integral with appropriate choices of a finite-dimensional function space that approximate sections of the configuratio...