Maximum error estimates of a MAC scheme for Stokes equations with Dirichlet boundary conditions

2020 ◽  
Vol 150 ◽  
pp. 149-163
Author(s):  
Haixia Dong ◽  
Wenjun Ying ◽  
Jiwei Zhang
Keyword(s):  
Boundary Conditions ◽  
Error Estimates ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Maximum Error ◽  
Dirichlet Boundary Conditions ◽  
Mac Scheme

Vertical Layers

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Point Of View ◽  
Dirichlet Boundary Conditions ◽  
Careful Study ◽  
Navier Stokes Equations ◽  
Mathematical Point ◽  
Open Questions ◽  
Ekman Layers

From a physical point of view, as well as from a mathematical point of view, horizontal layers (Ekman layers) are now well understood. This is not the casefor vertical layers which are much more complicated, from a physical, analytical and mathematical point of view, and many open questions in all these directions remain open. Let us, in this section, consider a domain Ω with vertical boundaries. Namely, let Ωh be a domain of R2 and let Ω=Ωh × [0, 1]. This domain has two types of boundaries: • horizontal boundaries Ωh × {0} (bottom) and Ωh ×{1} (top) where Ekman layers are designed to enforce Dirichlet boundary conditions; • vertical boundaries ∂Ωh × [0, 1] where again a boundary layer is needed to ensure Dirichlet boundary conditions. These layers, however, are not of Ekman type, since r is now parallel to the boundary. Vertical layers are quite complicated. They in fact split into two sublayers: one of size E1/3 and another of size E1/4 where E =νε denotes the Ekman number. This was discovered and studied analytically by Stewartson and Proudman. Vertical layers can be easily observed in experiments (at least the E1/4 layer, the second one being too thin) but do not seem to be relevant in meteorology or oceanography, where near continents, effects of shores, density stratification, temperature, salinity, or simply topography are overwhelming and completely mistreated by rotating Navier–Stokes equations. In MHD, however, and in particular in the case of rotating concentric spheres, they are much more important. Numerically, they are easily observed, at large Ekman numbers E (small Ekman numbers being much more difficult to obtain). The aim of this section is to provide an introduction to the study of these layers, a study mainly open from a mathematical point of view. First we will derive the equation of the E1/3 layer. Second we will investigate the E1/4 layer and underline its similarity with Prandtl’s equations. In particular, we conjecture that E1/4 is always linearly and nonlinearly unstable. We will not prove this latter fact, which would require careful study of what happens at the corners of the domain, a widely open problem.

scholarly journals MIXED FINITE ELEMENT APPROXIMATION OF THE VECTOR LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250024 ◽  
Author(s):  
DOUGLAS N. ARNOLD ◽  
RICHARD S. FALK ◽  
JAY GOPALAKRISHNAN
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Mixed Finite Element Method ◽  
Two Dimensions ◽  
Dirichlet Boundary Conditions ◽  
Element Approximation

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

Optimal-rate finite-element solution of Dirichlet problems in curved domains with straight-edged tetrahedra

2020 ◽  
Author(s):  
Vitoriano Ruas
Keyword(s):  
Boundary Conditions ◽  
Error Estimates ◽  
Degrees Of Freedom ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Dirichlet Problems ◽  
Simple Alternative ◽  
Curved Domains

Abstract In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

scholarly journals Charge algebra in Al(A)dSn spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Adrien Fiorucci ◽  
Romain Ruzziconi
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Structure ◽  
Asymptotic Symmetry ◽  
Renormalization Procedure ◽  
Field Dependent ◽  
Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

scholarly journals Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions

2021 ◽  
pp. 104123
Author(s):  
Firdous A. Shah ◽  
Mohd Irfan ◽  
Kottakkaran S. Nisar ◽  
R.T. Matoog ◽  
Emad E. Mahmoud
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Wavelet Method ◽  
Telegraph Equations
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