In this paper,we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem.Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme ...In this paper,we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem.Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be flexibly applied to fairly general polygonal meshes.We relax the tangential continuity for velocity,which is the key ingredi-ent in achieving the uniform robustness.We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme,and the theories indicate that the error estimates for velocity are independent of pressure.Several numerical experiments are presented to confirm the theoretical findings.展开更多
In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy＇s law and Stokes equations in a single form of PDE but with strongly disconti...In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy＇s law and Stokes equations in a single form of PDE but with strongly discontinuous viscosity coefficient and zerothorder term coefficient. We present three different methods to construct uniformly stable finite element approximations. The first two methods are based on the original weak formulations of Darcy-Stokes-Brinkman equations. In the first method we consider the existing Stokes elements. We show that a stable Stokes element is also uniformly stable with respect to the coefficients and the jumps of Darcy-Stokes-Brinkman equations if and only if the discretely divergence-free velocity implies almost everywhere divergence-free one. In the second method we construct uniformly stable elements by modifying some well-known H（div）-conforming elements. We give some new 2D and 3D elements in a unified way. In the last method we modify the original weak formulation of Darcy-Stokes- Brinkman equations with a stabilization term. We show that all traditional stable Stokes elements are uniformly stable with respect to the coefficients and their jumps under this new formulation.展开更多
The problem of uniform dimensions for multi-parameter processes, which may not possess the uniform stochastic H?lder condition, is investigated. The problem of uniform dimension for multi-parameter stable processes is...The problem of uniform dimensions for multi-parameter processes, which may not possess the uniform stochastic H?lder condition, is investigated. The problem of uniform dimension for multi-parameter stable processes is solved. That is, ifZ is a stable (N,d, α)-process and αN ?d, then $$\forall E \subseteq \mathbb{R}_ + ^N , \dim Z\left( E \right) = \alpha \cdot \dim E$$ holds with probability 1, whereZ(E) = {x : ?t ∈E,Z t =x} is the image set ofZ onE. The uniform upper bounds for multi-parameter processes with independent increments under general conditions are also given. Most conclusions about uniform dimension can be considered as special cases of our results.展开更多
基金the Hong Kong RGC General Research Fund(Project numbers 14304719 and 14302018)CUHK Faculty of Science Direct Grant 2019-20。
文摘In this paper,we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem.Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be flexibly applied to fairly general polygonal meshes.We relax the tangential continuity for velocity,which is the key ingredi-ent in achieving the uniform robustness.We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme,and the theories indicate that the error estimates for velocity are independent of pressure.Several numerical experiments are presented to confirm the theoretical findings.
基金NSF DMS-0609727by the Center for Computational Mathematics and Applications of Penn State+3 种基金Jinchao Xu was also supported in part by NSFC-10501001Alexander H.Humboldt Foundation.Xiaoping Xie was supported by the National Natural Science Foundation of China (10771150)the National Basic Research Program of China (2005CB321701)the program for New Century Excellent Talents in University (NCET-07-0584)
文摘In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy＇s law and Stokes equations in a single form of PDE but with strongly discontinuous viscosity coefficient and zerothorder term coefficient. We present three different methods to construct uniformly stable finite element approximations. The first two methods are based on the original weak formulations of Darcy-Stokes-Brinkman equations. In the first method we consider the existing Stokes elements. We show that a stable Stokes element is also uniformly stable with respect to the coefficients and the jumps of Darcy-Stokes-Brinkman equations if and only if the discretely divergence-free velocity implies almost everywhere divergence-free one. In the second method we construct uniformly stable elements by modifying some well-known H（div）-conforming elements. We give some new 2D and 3D elements in a unified way. In the last method we modify the original weak formulation of Darcy-Stokes- Brinkman equations with a stabilization term. We show that all traditional stable Stokes elements are uniformly stable with respect to the coefficients and their jumps under this new formulation.
基金Project supported by Fujian Natural Science Foundation.
文摘The problem of uniform dimensions for multi-parameter processes, which may not possess the uniform stochastic H?lder condition, is investigated. The problem of uniform dimension for multi-parameter stable processes is solved. That is, ifZ is a stable (N,d, α)-process and αN ?d, then $$\forall E \subseteq \mathbb{R}_ + ^N , \dim Z\left( E \right) = \alpha \cdot \dim E$$ holds with probability 1, whereZ(E) = {x : ?t ∈E,Z t =x} is the image set ofZ onE. The uniform upper bounds for multi-parameter processes with independent increments under general conditions are also given. Most conclusions about uniform dimension can be considered as special cases of our results.
