Hybrid High-Order (HHO) methods are formulated in terms of discrete unknowns attached to mesh faces and cells ( hence, the term hybrid), and these unknowns are polynomials of arbitrary order k >= 0 ( hence, the ter...
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ISBN:
(纸本)9783319416403;9783319416380
Hybrid High-Order (HHO) methods are formulated in terms of discrete unknowns attached to mesh faces and cells ( hence, the term hybrid), and these unknowns are polynomials of arbitrary order k >= 0 ( hence, the term high-order). HHO methods are devised from local reconstruction operators and a local stabilization term. The discrete problem is assembled cellwise, and cell-based unknowns can be eliminated locally by static condensation. HHO methods support generalmeshes, are locally conservative, and allowfor a robust treatment of physical parameters in various situations, e.g., heterogeneous/anisotropic diffusion, quasi-incompressible linear elasticity, and advection-dominated transport. This paper reviews HHO methods for a variable-diffusion model problem with nonhomogeneous, mixed Dirichlet-Neumann boundary conditions, including both primal and mixed formulations. Links with other discretization methods from the literature are discussed.
We present a parallel and efficient multilevel solution method for the nonlinear systems arising from the discretization of Navier-Stokes (N-S) equations with finite differences. In particular we study the incompressi...
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ISBN:
(纸本)9783319399294;9783319399270
We present a parallel and efficient multilevel solution method for the nonlinear systems arising from the discretization of Navier-Stokes (N-S) equations with finite differences. In particular we study the incompressible, unsteady N-S equations with periodic boundary condition in time. A sequential time integration limits the parallelism of the solver to the spatial variables and can therefore be an obstacle to parallel scalability. Time periodicity allows for a space-time discretization, which adds time as an additional direction for parallelism and thus can improve parallel scalability. To achieve fast convergence, we used a spacetime multigrid algorithm with a SCGS smoothing procedure (symmetrical coupled Gauss-Seidel, a. k. a. box smoothing). This technique, proposed by Vanka (J Comput Phys 65: 138-156, 1986), for the steady viscous incompressible Navier-Stokes equations is extended to the unsteady case and its properties are studied using local Fourier analysis. We used numerical experiments to analyze the scalability and the convergence of the solver, focusing on the case of a pulsatile flow.
An abstract setting for the construction and analysis of the Multiscale Hybrid-Mixed (MHM for short) method is proposed. We review some of the most recent developments from this standpoint, and establish relationships...
ISBN:
(纸本)9783319416403;9783319416380
An abstract setting for the construction and analysis of the Multiscale Hybrid-Mixed (MHM for short) method is proposed. We review some of the most recent developments from this standpoint, and establish relationships with the classical lowest-order Raviart-Thomas element and the primal hybrid method, as well as with some recent multiscale methods. We demonstrate the reach of the approach by revisiting the wellposedness and error analysis of the MHM method applied to the Laplace problem. In the process, we devise new theoretical results for this model.
This paper is concerned with the numerical solution of dynamic elasticity by the discontinuous Galerkin (dG) method. We consider the linear and nonlinear St. Venant-Kirchhoff model. The dynamic elasticity problem is s...
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ISBN:
(纸本)9783319399294;9783319399270
This paper is concerned with the numerical solution of dynamic elasticity by the discontinuous Galerkin (dG) method. We consider the linear and nonlinear St. Venant-Kirchhoff model. The dynamic elasticity problem is split into two systems of first order in time. They are discretized by the discontinuous Galerkin method in space and backward difference formula in time. The developed method is tested by numerical experiments. Then the method is combined with the space-time dG method for the solution of compressible flow in a time dependent domain and used for the numerical simulation of fluid-structure interaction.
We are interested in the numerical solution of linear hyperbolic problems using continuous finite elements of arbitrary order. It is well known that this kind of methods, once the weak formulation has been written, le...
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ISBN:
(纸本)9783319399294;9783319399270
We are interested in the numerical solution of linear hyperbolic problems using continuous finite elements of arbitrary order. It is well known that this kind of methods, once the weak formulation has been written, leads to a system of ordinary differential equations in RN, where N is the number of degrees of freedom. The solution of the resulting ODE system involves the inversion of a sparse mass matrix that is not block diagonal. Here we show how to avoid this step, and what are the consequences of the choice of the finite element space. Numerical examples show the correctness of our approach.
The purpose of the paper is to analyse the effect of hp mesh adaptation when discretized versions of finite elementmixed formulations are applied to elliptic problems with singular solutions. Two stable configurations...
ISBN:
(纸本)9783319399294;9783319399270
The purpose of the paper is to analyse the effect of hp mesh adaptation when discretized versions of finite elementmixed formulations are applied to elliptic problems with singular solutions. Two stable configurations of approximation spaces, based on affine triangular and quadrilateral meshes, are considered for primal and dual (flux) variables. When computing sufficiently smooth solutions using regular meshes, the first configuration gives optimal convergence rates of identical approximation orders for both variables, as well as for the divergence of the flux. For the second configuration, higher convergence rates are obtained for the primal variable. Furthermore, after static condensation is applied, the condensed systems to be solved have the same dimension in both configuration cases, which is proportional to their border flux dimensions. A test problem with a steep interior layer is simulated, and the results demonstrate exponential rates of convergence. Comparison of the results obtained with H-1-conforming formulation are also presented.
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