Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations

In this paper we present in a unified setting the continuous and discontinuous Galerkin methods for the numerical approximation of the scalar hyperbolic equation. Both methods are stabilized by the interior penalty method, more precisely by the jump of the gradient across element faces in the continuous case whereas in the discontinuous case the stabilization of the jump of the solution and optionally of its gradient is required to achieve optimal convergence. We prove that the solution in the case of the continuous Galerkin approach can be considered as a limit of the discontinuous one when the stabilization parameter associated with the penalization of the solution jump tends to infinity. As a consequence, the limit of the numerical flux of the discontinuous method yields a numerical flux for the continuous method as well. Numerical results will highlight the theoretical results that are proven in this pape

High order discontinuous Galerkin methods on surfaces

We derive and analyze high order discontinuous Galerkin methods for second-order elliptic problems on implicitely defined surfaces in $\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous Galerkin framework of Arnold et al. [2002] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy norm and the $L^2$ norm.Comment: 23 pages, 2 figure

The a posteriori error estimates of the Ciarlet-Raviart mixed finite element method for the biharmonic eigenvalue problem

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics and has wide applications. In this paper, for the biharmonic eigenvalue problem, based on the work of Gudi [Numer. Methods Partial Differ. Equ., 27 (2011), 315-328], we study the a posteriori error estimates of the approximate eigenpairs obtained by the Ciarlet-Raviart mixed finite element method. We prove the reliability and efficiency of the error estimator of the approximate eigenfunction and analyze the reliability of the error estimator of the approximate eigenvalues. We also implement the adaptive calculation and exhibit the numerical experiments which show that our method is efficient and can get an approximate solution with high accuracy

Discontinuous subgrid formulations for transport problems

In this paper we develop two discontinuous Galerkin formulations within the framework of the two-scale subgrid method for solving advection–diffusion-reaction equations. We reformulate, using broken spaces, the nonlinear subgrid scale (NSGS) finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh. Here, two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges. The amount of subgrid viscosity is scaled by the resolved scale solution at the element level, yielding a self adaptive method so that no additional stabilization parameter is required. Numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous Galerkin methods.Indisponível

Convergence of adaptive discontinuous Galerkin and C⁰-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations

We prove the convergence of adaptive discontinuous Galerkin and C0-interior penalty methods for fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We consider a broad family of methods on adaptively refined conforming simplicial meshes in two and three space dimensions, with fixed but arbitrary polynomial degrees greater than or equal to two. A key ingredient of our approach is a novel intrinsic characterization of the limit space that enables us to identify the weak limits of bounded sequences of nonconforming finite element functions. We provide a detailed theory for the limit space, and also some original auxiliary functions spaces, that is of independent interest to adaptive nonconforming methods for more general problems, including Poincaré and trace inequalities, a proof of the density of functions with nonvanishing jumps on only finitely many faces of the limit skeleton, approximation results by finite element functions and weak convergence results

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications