排序方式: 共有26条查询结果,搜索用时 15 毫秒
1.
Markus Hegland Jochen Garcke Vivien Challis 《Linear algebra and its applications》2007,420(2-3):249-275
The combination technique has repeatedly been shown to be an effective tool for the approximation with sparse grid spaces. Little is known about the reasons of this effectiveness and in some cases the combination technique can even break down. It is known, however, that the combination technique produces an exact result in the case of a projection into a sparse grid space if the involved partial projections commute.
The performance of the combination technique is analysed using a projection framework and the C/S decomposition. Error bounds are given in terms of angles between the spanning subspaces or the projections onto these subspaces. Based on this analysis modified combination coefficients are derived which are optimal in a certain sense and which can substantially extend the applicability and performance of the combination technique. 相似文献
2.
Weizhu Bao Harald Garcke Robert Nürnberg Quan Zhao 《Numerical Methods for Partial Differential Equations》2023,39(1):759-794
We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points lines, where three interfaces meet, and at the boundary points lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies. 相似文献
3.
John?W.?Barrett Harald?Garcke Robert?NürnbergEmail author 《Numerische Mathematik》2012,120(3):489-542
Deckelnick and Dziuk (Math. Comput. 78(266):645–671, 2009) proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of the elastic flow
of closed curves in
\mathbbRd, d 3 2{\mathbb{R}^d, d\geq2} . We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of
the features of the formulations in Barrett et al. (J Comput Phys 222(1): 441–462, 2007; SIAM J Sci Comput 31(1):225–253, 2008; IMA J Numer Anal 30(1):4–60, 2010), in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time
semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition
to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the
plane and in higher codimension that satisfies various boundary conditions. 相似文献
4.
We consider the sharp interface limit of a diffuse phase-field model with prescribed total mass taking into account a spatially inhomogeneous, anisotropic interfacial energy and an elastic energy. The main aim is to establish a weak formulation of an inhomogeneous, anisotropic, elastically modified Gibbs-Thomson law in the sharp interface limit. To this end we pass to the limit in the weak formulation of the Euler-Lagrange equation of the diffuse phase-field energy. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
5.
Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn–Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn–Hilliard–Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting. 相似文献
6.
We present a novel variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow in , d ≥ 2. This new formulation leads to an unconditionally stable, fully discrete, parametric finite element approximation in
the case d = 2 or 3. The resulting scheme has very good properties with respect to the distribution of mesh points and, if applicable,
volume conservation. This is demonstrated by several numerical experiments for d = 3, including regularized crystalline mean curvature flow and regularized crystalline surface diffusion. 相似文献
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8.
Barrett John W.; Garcke Harald; Nurnberg Robert 《IMA Journal of Numerical Analysis》2008,28(2):292-330
9.
Harald Garcke Michael Hinze Christian Kahle Kei Fong Lam 《Advances in Computational Mathematics》2018,44(5):1345-1383
We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization. 相似文献
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