1,721,136 research outputs found
Hierarchical model reduction for advection-diffusion-reaction problems
No full textK. Kunisch, G. Of, O. Steinbach Eds
Some Variants
No full textThe goal of this chapter is to explore some variants of the HHO method devised in Chap. 1 and analyzed in Chap. 2. We first study two variants of the gradient reconstruction operator that will turn useful, for instance, when dealing with nonlinear problems in Chaps. 4 and 7. Then, we explore a mixed-order variant of the HHO method that is useful, for instance, to treat domains with a curved boundary. Finally, we bridge the HHO method to the finite element and virtual element viewpoints
Mathematical Aspects
No full textThe objective of this chapter is to put the HHO method presented in the previous chapter on a firm mathematical ground. In particular, we prove the key stability and convergence results announced in the previous chapter
Getting Started: Linear Diffusion
No full textThe objective of this chapter is to gently introduce the hybrid high-order (HHO) method on one of the simplest model problems: the Poisson problem with homogeneous Dirichlet boundary conditions. Our goal is to present the key ideas underlying the devising of the method and state its main properties (most of them without proof). The keywords of this chapter are cell and face unknowns, local reconstruction and stabilization operators, elementwise assembly, static condensation, energy minimization, and equilibrated fluxes
Plasticity
No full textModeling plasticity problems is particularly relevant in nonlinear solid mechanics since plasticity can have a major influence on the behavior of a mechanical structure. One difficulty is that the plastic deformations are generally assumed to be incompressible, leading to volume-locking problems if (low-order) H1 -conforming finite elements are used. Mixed methods avoid these problems, but need additional globally coupled unknowns to enforce the incompressibility of the plastic deformations
Implementation Aspects
No full textIn this chapter, we outline the steps needed to bring the abstract formulation of the HHO method to an actual implementation. For simplicity, we focus on the Poisson model problem (see Chap. 1 ). We show how the local HHO operators (reconstruction and stabilization) are translated into matrices that can be used in the actual computation, and we give some criteria to test the implementation. Then we discuss the assembly of the discrete problem and the handling of the boundary conditions. We conclude with a brief overview on computational costs. Along the chapter, we provide some snippets of Matlab®/Octave code to show a possible implementation (in 1D) of the critical parts
A Discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity
No full textWe 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 P ́eclet numbers) on tted
meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect
to mesh-size and robust with respect to diffusivity. The convergence results for the advective derivative
are optimal with respect to mesh-size and robust for isotropic diffusivity, as well as for anisotropic diffusivity
if the cell P ́eclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large
enough. Numerical results are presented to illustrate the performance of the proposed scheme
Linear Elasticity and Hyperelasticity
No full textIn this chapter, we show how to discretize using HHO methods linear elasticity and nonlinear hyperelasticity problems. In particular, we pay particular attention to the robustness of the discretization in the quasi-incompressible limit. For linear elasticity, we reconstruct the strain tensor in the space composed of symmetric gradients of vector-valued polynomials. For nonlinear hyperelasticity, we reconstruct the deformation gradient in a full tensor-valued polynomial space, and not just in a space composed of polynomial gradients. We also consider a second gradient reconstruction in an even larger space built using Raviart–Thomas polynomials, for which no additional stabilization is necessary. Finally, we present some numerical examples
Hybrid High-Order Methods for the Elliptic Obstacle Problem
No full textHybrid High-Order methods are introduced and analyzed for the elliptic obstacle problem in two and three space dimensions. The methods are formulated in terms of face unknowns which are polynomials of degree k= 0 or k= 1 and in terms of cell unknowns which are polynomials of degree l= 0. The discrete obstacle constraints are enforced on the cell unknowns. Higher polynomial degrees are not considered owing to the modest regularity of the exact solution. A priori error estimates of optimal order, that is, up to the expected regularity of the exact solution, are shown. Specifically, for k= 1 , the method employs a local quadratic reconstruction operator and achieves an energy-error estimate of order h32-ε, ε> 0. To our knowledge, this result fills a gap in the literature for the quadratic approximation of the three-dimensional obstacle problem. Numerical experiments in two and three space dimensions illustrate the theoretical results
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