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18
field games with nonlinear mobilities in pedestrian dynamics, submitted
, 2013
"... Abstract. In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which ..."
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Abstract. In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which depends on his/her position, velocity, exit time and the overall density of people. This microscopic setup leads in the meanfield limit to a parabolic optimal control problem. We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate to Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers and illustrate the behavior of the model with various numerical results. 1.
A HYBRID MORTAR FINITE ELEMENT METHOD FOR THE STOKES PROBLEM
, 2010
"... In this paper, we consider the discretization of the Stokes problem on domain partitions with nonmatching meshes. We propose a hybrid mortar method, which is motivated by a variational characterization of solutions of the corresponding interface problem. For the discretization of the subdomain pro ..."
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Cited by 5 (2 self)
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In this paper, we consider the discretization of the Stokes problem on domain partitions with nonmatching meshes. We propose a hybrid mortar method, which is motivated by a variational characterization of solutions of the corresponding interface problem. For the discretization of the subdomain problems, we utilize standard infsup stable finite element pairs. The introduction of additional unkowns at the interface allows to reduce the coupling between the subdomain problems, which comes from the variational incorporation of interface conditions. We present a detailed analysis of the hybrid mortar method, in particular, the discrete infsup stability condition is proven under weak assumptions on the interface mesh, and optimal apriori error estimates are derived with respect to the energy and L²norm. For illustration of the results, we present some numerical tests.
A Hybridized DG/Mixed Scheme for Nonlinear AdvectionDiffusion Systems, Including the Compressible NavierStokes Equations. AIAA Paper
, 2012
"... We present a novel discretization method for nonlinear convectiondiffusion equations and, in particular, for the compressible NavierStokes equations. The method is based on a Discontinuous Galerkin (DG) discretization for convection terms, and a mixed method using H(div) spaces for the diffusive t ..."
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Cited by 3 (2 self)
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We present a novel discretization method for nonlinear convectiondiffusion equations and, in particular, for the compressible NavierStokes equations. The method is based on a Discontinuous Galerkin (DG) discretization for convection terms, and a mixed method using H(div) spaces for the diffusive terms. Furthermore, hybridization is used to reduce the number of globally coupled degrees of freedom. The method reduces to a DG scheme for pure convection, and to a mixed method for pure diffusion, while for the intermediate case the combined variational formulation requires no additional parameters. We formulate and validate our scheme for nonlinear model problems, as well as compressible flow problems. Furthermore, we compare our scheme to a recently developed Hybridized DG scheme with respect to formulation and convergence behavior. I.
AN ANALYSIS OF HDG METHODS FOR CONVECTION–DOMINATED DIFFUSION PROBLEMS
"... Abstract. We present the first a priori error analysis of the h–version of the hybridizable discontinuous Galkerin (HDG) methods applied to convection–dominated diffusion problems. We show that, when using polynomials of degree no greater than k, the L2–error of the scalar variable converges with o ..."
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Abstract. We present the first a priori error analysis of the h–version of the hybridizable discontinuous Galkerin (HDG) methods applied to convection–dominated diffusion problems. We show that, when using polynomials of degree no greater than k, the L2–error of the scalar variable converges with order k + 1/2 on general conforming quasi–uniform simplicial meshes, just as for conventional DG methods. We also show that the method achieves the optimal L2–convergence order of k+ 1 on special meshes. Moreover, we discuss a new way of implementing the HDG methods for which the spectral condition number of the global matrix is independent of the diffusion coefficient. Numerical experiments are presented which verify our theoretical results. 1.
Adjointbased error estimation and mesh adaptation for hybridized discontinuous Galerkin methods
, 2013
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"... Abstract. A hybrid method for the incompressible Navier–Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equalorder interpolations can be used for the velocity and pressur ..."
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Abstract. A hybrid method for the incompressible Navier–Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equalorder interpolations can be used for the velocity and pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh. Different from our earlier investigations on the approach for the Navier–Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for an appropriate choice of finite element spaces, momentum conservation. Also, a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a pointwise solenoidal velocity field. Mass conservation, momentum conservation, and global energy stability are proved for the timecontinuous case and for a fully discrete scheme. The presented analysis results are supported by a range of numerical simulations.
pp. X–XX A COMBINED HYBRIDIZED DISCONTINUOUS GALERKIN / HYBRID MIXED METHOD FOR VISCOUS CONSERVATION
"... hybridized discontinuous Galerkin method, hybrid mixed method, viscous conservation laws, time–discretization, backward difference schemes. Mathematics Subject Classifications: ..."
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hybridized discontinuous Galerkin method, hybrid mixed method, viscous conservation laws, time–discretization, backward difference schemes. Mathematics Subject Classifications:
Decomposition Method
, 2011
"... Abstract In this paper we present a hybrid domain decomposition approach for the parallel solution of linear systems arising from a discontinuous Galerkin (DG) finite element approximation of initial boundary value problems. This approach allows a general decomposition of the space–time cylinder int ..."
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Abstract In this paper we present a hybrid domain decomposition approach for the parallel solution of linear systems arising from a discontinuous Galerkin (DG) finite element approximation of initial boundary value problems. This approach allows a general decomposition of the space–time cylinder into finite elements, and is therefore applicable for adaptive refinements in space and time. 1 A Space–Time DG Finite Element Method As a model problem we consider the transient heat equation ∂tu(x,t)−∆u(x,t) = f(x,t) for (x,t) ∈ Q: = Ω ×(0,T), (1) u(x,t) = 0 for (x,t) ∈ Σ: = ∂Ω ×(0,T), (2) u(x,0) = u0(x) for (x,t) ∈ Ω ×{0} (3) where Ω ⊂ R n,n = 1,2,3, is a bounded Lipschitz domain, and T> 0. Let TN be a decomposition of the space–time cylinder Q = Ω × (0,T) ⊂ R n+1 into simplices τk of mesh size h. For simplicity we assume that the space time cylinder Q has a polygonal (n = 1), a polyhedral (n = 2), or a polychoral (n = 3) boundary ∂Q. With IN we denote the set of all interfaces (interior elements) e between two neighbouring elements τk and τℓ. For an admissible decomposition the interior elements are edges (n = 1), triangles (n = 2), or tetrahedrons (n = 3). With respect to an interior element e ∈ IN we define for a function v the jump
for parabolic initial boundary value problems
, 2011
"... A flexible spacetime discontinuous Galerkin method for parabolic initial boundary value problems ..."
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A flexible spacetime discontinuous Galerkin method for parabolic initial boundary value problems