Method Of Asymptotic Partial Decomposition Research Articles

Method of asymptotic partial decomposition with discontinuous junctions

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the heat equation these conditions ensure the continuity of the solution and continuity in average of the normal flux. However for some computational reasons the strong continuity condition for the solution may be undesirable. It may be preferable to replace it by more flexible condition of continuity in average over the interface cross section. In the present paper we introduce and justify this alternative junction condition allowing such discontinuity of the solution of both stationary and non-stationary problems. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. At the discrete level, finite volume schemes are considered, an error estimate is established. The new version of the MAPDD is compared to the classical one via numerical tests. It shows better stability of the new scheme.

Reconstruction of the Pressure in the Method of Asymptotic Partial Decomposition for the Flows in Tube Structures

The method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates ...

Time periodic Navier\u2013Stokes equations in a thin tube structure

The time periodic Navier–Stokes equations are considered in the three-dimensional and two-dimensional settings with Dirichlet boundary conditions in thin tube structures. These structures are finite union of thin cylinders (thin rectangles in the case of dimension two), where the small parameter ε is the ratio of the hight and the diameter of the cylinders. We consider the case of finite or big coefficient before the time derivative. This setting is motivated by hemodynamical applications. Theorems of existence and uniqueness of a solution are proved. Complete asymptotic expansion of a solution is constructed and justified by estimates of the difference of the exact solution and truncated series of the expansion in norms taking into account the first and second derivatives with respect to the space variables and the first derivative in time. The method of asymptotic partial decomposition of the domain is justified for the time periodic problem.

Junction of Models of Different Dimension for Flows in Tube Structures by Womersley-Type Interface Conditions

The method of asymptotic partial decomposition of a domain proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some spe...

Partial dimension reduction for the heat equation in a domain containing thin tubes

The heat equation in considered in a domain containing thin cylindrical tubes with Neumann's boundary condition at the lateral boundary of these tubes. This problem is reduced to a hybrid dimension problem keeping the initial dimension out of thin tubes and reducing it to the one‐dimensional heat equation within the tubes at some distance from the bases of the cylinders. Junction of models of different dimensions is done according to the method of asymptotic partial decomposition of domain. The difference of solutions of the original and partially reduced problems is estimated. This result generalizes the method of partial dimension reduction for the case when the domain has only one restriction that it contains several thin cylinders possibly of different orders of diameters.

Method of asymptotic partial decomposition of domain for multistructures

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures) is generalized and justified for the multistructures, i.e. domains consisting of a set of thin cylinders connecting some massive 3D domains. In the present paper, the Dirichlet boundary value problem for the steady-state Stokes equations is considered. This problem is reduced to the Stokes equations in the massive domains coupled with the Poiseuille-type flows within the thin cylinders at some distance from the bases (the MAPDD approximation problem). The high-order estimates for the difference of the exact solution to the initial problem and the solution to the MAPDD approximation problem is proved.

A posteriori estimate and asymptotic partial domain decomposition

The method of asymptotic partial decomposition of a domain aims at replacing a 3D or 2D problem by a hybrid problem 3D − 1D; or 2D − 1D, where the dimension of the problem decreases in part of the domain. The location of the junction between the heterogeneous problems is asymptotically estimated in certain circumstances, but for numerical simulations it is important to be able to determine the location of the junction accurately. In this article, by reformulating the problem in a mixed formulation context and by using an a posteriori error estimate, we propose an indicator of the error due to a wrong position of the junction. Minimizing this indicator allows us to determine accurately the location of the junction. Some numerical results are presented for a toy problem.

Asymptotic Domain Decomposition and a Posteriori Estimates for a Semi Linear Problem

Keywords: Method of asymptotic partial domain decomposition; a posteriori error estimates; Indicator of the error; semi linear elliptic equations. 2010 Mathematics Subject Classification: 35F40; 65 1 Introduction The method of asymptotic partial decomposition of a domain (MAPDD) originates with the works of G.Panasenko Panasenko (2005). The idea is to replace an original 3D or 2D problem by an hybrid one 3D − 1D; or 2D − 1D where the dimension of the problem decreases in part of the domain. The location of the junction between the heterogenous problems is asymptotically estimated in the works of G.Panasenko mainly for linear problems. Nevertheless for numerical simulations it is essential to detect with accuracy the location of the junction. Let us also mention the interest of locating with accuracy the position of the junction in blood flows simulations when different nonlinear mathematical models are used Quarteroni and Veneziani (2003), or in fluid/solid problems for which subproblems *Corresponding author: Tel:+91-9741148002, E-mail: jerome.pousin@insa-lyon.fr Research Article Coupling heterogeneous mathematical models is today commonly used, and effective solution methods for the resulting hybrid problem have recently become available for several systems. Even if in certain circumstances, asymptotic evaluations of the location of the interfaces are available, no strategy are proposed for locating the interfaces in numerical simulations. In this article, a semi- linear elliptic problem is considered. By reformulating the problem in a mixed formulation context and by using an a posteriori error estimate, we propose an indicator of the error due to a wrong position of the junction. Minimizing this indicator allows us to determine accurately the location of the junction. By comparing this indicator with a mesh error indicator, this allows to decide if it is better to refine the mesh or to move the interface. Some numerical results are presented showing the efficiency of the proposed indicator.

THE PARTIAL HOMOGENIZATION: CONTINUOUS AND SEMI-DISCRETIZED VERSIONS

The partial homogenization is a new method for the treatment of the boundary layers in the homogenization theory. It keeps the initial formulation near the boundary, passes to the high order homogenization at some distance from the boundary and prescribes the asymptotically precise junction conditions between the homogenized and the heterogeneous models at the interface. This method is related to the method of asymptotic partial domain decomposition MAPDD (see G. Panasenko, Method of asymptotic partial decomposition of domain, Math. Mod. Meth. Appl. Sci.8 (1998) 139–156). The partial homogenization (as well as the MAPDD) can be interpreted as a multi-scale model coupling the homogenized (macroscopic) description in the internal main part of the domain and the microscopic zoom in the domain of the location of the boundary layers. The semi-discretized partial homogenization uses some high order finite element projection in the homogenized subdomain.

Asymptotic analysis of integral equations for a great interval and its application to stellar radiative transfer

We consider the integral form of the radiative transfer equation over a large interval. This equation describes the radiative transfer of energy in a star. The asymptotic expansion of the solution is constructed and justified. The method of asymptotic partial decomposition of domain is applied. Numerical results are discussed. To cite this article: G. Panasenko et al., C. R. Mecanique 330 (2002) 735–740.

ASYMPTOTIC ANALYSIS AND PARTIAL ASYMPTOTIC DECOMPOSITION OF DOMAIN FOR STOKES EQUATION IN TUBE STRUCTURE

The Stokes problem posed in tube structures (or finite rod structures (see Panasenko10)), i.e. in connected finite unions of the thin cylinders with the ratio of the diameter to the height of the order [Formula: see text], is considered. The asymptotic expansion of the solution is built and justified. Boundary layers are studied. Earlier the Navier–Stokes problem in one thin domain was considered by Nazarov.8 The method of asymptotic partial decomposition of the domain (MAPDD) (see Panasenko11) is applied and justified for the Stokes problem posed in a tube structures. This method reduces the initial Stokes problem to the Stokes problem in some small parts of the domain (where the boundary layers are "concentrated".)

Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure

The Navier-Stokes problem stated in tube structures (or finite rod structures), i.e. in connected finite unions of thin cylinders with ratio of the diameter to the height of the order ε < < 1, is considered. The method of asymptotic partial decomposition of the domain (MAPDD) is applied and justified for the Navier-Stokes problem posed in a tube structure. This method reduces the initial Navier-Stokes problem to the Navier-Stokes problem in certain small parts of the domain (where the boundary layers are “concentrated”).