Browsing by Author "Cockburn, B."
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Item Continuous Dependence on the Nonlinearities of Solutions of Degenerate Parabolic Equations(1997-08) Cockburn, B.; Gripenberg, G.Item Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations(1999-12) Cockburn, B.; Gripenberg, G.; Londen, S-O.Item Convergence of finite volume methods(1991-02) Cockburn, B.; Coquel, F.; LeFloch, Ph.; Shu, C.W.Item Determining degrees of freedom for nonlinear dissipative equations(1995-05) Cockburn, B.; Jones, D.A.; Titi, E.S.Item Discontinuous Galerkin methods(2003-05) Cockburn, B.This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approximations, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to non-linear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution. In this paper, we concentrate on the exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems.Item Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems(1995-08) Cockburn, B.; Jones, D.A.; Titi, E.S.Item The local discontinuous Galerkin method for time-dependent convection-diffusion systems(1997-03) Cockburn, B.; Shu, Chi-WangItem The Local Projection P0P1-Discontinuous-Galerkin-Finite Element Method for Scalar Conservation Laws(1987) Chavent, G.; Cockburn, B.Item On convergence to entropy solutions of a single conservation law(1994-12) Cockburn, B.; Gripenberg, G.; Londen, S-O.Item A posteriori error estimates for general numerical methods for scalar conservation laws(1994-07) Cockburn, B.; Gau, H.Item The Quasi-Monotone Schemes for Scalar Conservation Laws(1986) Cockburn, B.Item The Quasi-Monotone Schemes for Scalar Conservation Laws, Part III(1986) Cockburn, B.Item The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V: Multidimensional Systems(1997-06) Cockburn, B.; Shu, Chi-WangItem The TVD-Projection Method for Solving Implicit Numeric Schemes for Scalar Conservation Laws: A Numerical Study of a Simple Case(1987) Bourgeat, A.; Cockburn, B.Item Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems(2008-09-16) Cockburn, B.; Gopalakrishnan, J.; Lazarov, R.We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continu- ous Galerkin, non-conforming and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric and positive definite, these methods can be efficiently implemented. Moreover, the framework allows, in a single implementation, the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled. Finally, the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom.