The purpose of this paper is to study the asymptotic behavior of a viscous fluid satisfying Navier's condition on a slightly rough boundary. We consider the case of a fluid contained in a domain that has height I ...
详细信息
ISBN:
(数字)9783642196652
ISBN:
(纸本)9783642196645
The purpose of this paper is to study the asymptotic behavior of a viscous fluid satisfying Navier's condition on a slightly rough boundary. We consider the case of a fluid contained in a domain that has height I and the case of a fluid contained in a domain of small height e. In both cases we show that three different behaviors are possible.
The objective of this paper is to give an overview of the finite element approximation of the convection-diffusion equation that we have been developing in our group during the last years, together with some recent me...
详细信息
ISBN:
(数字)9783642196652
ISBN:
(纸本)9783642196645
The objective of this paper is to give an overview of the finite element approximation of the convection-diffusion equation that we have been developing in our group during the last years, together with some recent methods. We discuss three main aspects, namely, the global stabilization in the convective dominated regime, the treatment of the local instabilities that still remain close to layers when a stabilized formulation is used and the way to deal with transient problems. The starting point of our formulation is the variational multiscale framework. The main idea is to split the unknown into a finite element component and a remainder that is assumed that the finite element mesh cannot resolve. A closed form expression is then proposed for this remainder, referred to as subgrid-scale. When inserted into the equation for the finite element component, a method with enhanced stability properties is obtained. In our approach, we take the space for the subgrid-scales orthogonal to the finite element space. Once global instabilities have been overcome, there are still local oscillations near layers due to the lack of monotonicity of the method. Shock capturing techniques are often employed to deal with them. Here, our point of view is that this lack of monotonicity is inherent to the integral as duality pairing intrinsic to the variational formulation of the problem. We claim that if appropriate weighting functions are introduced when computing the integral, giving a reduced weight to layers, the numerical behavior of the method is greatly improved. The final point we treat is the time integration in time-dependent problems. Most stabilized finite element method require a link between the time step size of classical finite difference schemes in time and the mesh size employed for the spatial discretization. We show that this can be avoided by considering the subgrid-scales as time dependent, and discretizing them in time as well. That allows us to perform a complete n
The Stokes problem plays an important role in computational fluid dynamics since it is encountered in the time discretization of (incompressible) Navier-Stokes equations by operator-splitting methods [2, 3]. Space dis...
详细信息
In our earlier work [4], a dual iterative substructuring method for two dimensional problems was proposed, which is a variant of the FETI-DP method. The proposed method imposes continuity on the interface by not only ...
详细信息
ISBN:
(纸本)9783642113031
In our earlier work [4], a dual iterative substructuring method for two dimensional problems was proposed, which is a variant of the FETI-DP method. The proposed method imposes continuity on the interface by not only the pointwise matching condition but also uses a penalty term which measures the jump across the interface. For a large penalization parameter, it was proven that the condition number of the resultant dual problem is bounded by a constant independent of both the subdomain size H and the mesh size h. In this paper, we extend the method to three dimensional problems. For this extension, we consider two things;one is the construction of a penalty term in 3D to give the same convergence speed as in 2D and the other is how to treat the ill-conditioning of the subdomain problems due to a large penalization parameter. To resolve these two key issues, we need to be aware of the difference between 2D and 3D in the geometric complexity of the interface. Based on the geometric observation for the difference, we suggest a modified penalty term and a preconditioner aiming at reducing couplings between functions on the interface.
Optimized Schwarz methods form a class of domain decomposition methods for the solution of partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artifici...
详细信息
A singularly perturbed elliptic problem is considered on the unit square. Its boundary data has a jump discontinuity at one corner of the square, so the solution of the problem exhibits a singularity there. To solve t...
详细信息
ISBN:
(数字)9783642196652
ISBN:
(纸本)9783642196645
A singularly perturbed elliptic problem is considered on the unit square. Its boundary data has a jump discontinuity at one corner of the square, so the solution of the problem exhibits a singularity there. To solve the problem numerically, the Galerkin finite element method is tested on various tensor-product meshes. It is demonstrated that the Shishkin mesh does not yield satisfactory results, but meshes with a sufficient degree of mesh grading will yield convergence in certain norms, uniformly in the singular perturbation parameter.
Schwarz waveform relaxation methods are naturally parallel methods to solve evolution problems. They are based on a decomposition of the physical domain into overlapping subdomains, and a decomposition of the time dom...
详细信息
We present several coupled finite and boundary element formulations for the vibro-acoustic simulation of completely immersed bodies such as submarines. All formulations are based on the different use of standard bound...
详细信息
ISBN:
(纸本)9783642113031
We present several coupled finite and boundary element formulations for the vibro-acoustic simulation of completely immersed bodies such as submarines. All formulations are based on the different use of standard boundary integral equations. In addition to the well known symmetric coupling we discuss two different approaches which are based on the weakly singular boundary integral equation only.
We study the effect of different stabilized finite element methods to distributed control problems governed by singular perturbed Oseen equations. On the one hand, the residual based stabilized finite element method S...
详细信息
ISBN:
(数字)9783642196652
ISBN:
(纸本)9783642196645
We study the effect of different stabilized finite element methods to distributed control problems governed by singular perturbed Oseen equations. On the one hand, the residual based stabilized finite element method SUPG/PSPG leads to different optimality systems depending on the discretization approach: first discretize the state equation and then formulate the corresponding optimality system or derive first the optimality system on the continuous level and then discretize it. On the other hand, for symmetric stabilization as for instance the local projection stabilization (LPS) both approaches lead to the same symmetric optimality system. In particular, we address the question whether a possible commutation error in optimal control problems with boundary layers discretized by stabilized finite element methods may affect the accuracy significantly or not.
暂无评论