scholarly journals An Optimal Finite Element Method with Uzawa Iteration for Stokes Equations including Corner Singularities

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Jae-Hong Pyo ◽  
Deok-Kyu Jang
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Elliptic Boundary ◽  
Singular Function ◽  
Corner Singularities ◽  
Uzawa Method ◽  
Elliptic Boundary Value ◽  
Singular Functions ◽  
Pressure Variable ◽  
Lagrangian Operator

The Uzawa method is an iterative approach to find approximated solutions to the Stokes equations. This method solves velocity variables involving augmented Lagrangian operator and then updates pressure variable by Richardson update. In this paper, we construct a new version of the Uzawa method to find optimal numerical solutions of the Stokes equations including corner singularities. The proposed method is based on the dual singular function method which was developed for elliptic boundary value problems. We estimate the solvability of the proposed formulation and special orthogonality form for two singular functions. Numerical convergence tests are presented to verify our assertion.

scholarly journals On the Existence, Uniqueness and Application of the Finite Difference Method for Solving Robin Elliptic Boundary Value Problem

2019 ◽  
Vol 11 (1) ◽  
pp. 26
Author(s):  
Germain Nguimbi ◽  
Diogène Vianney Pongui Ngoma ◽  
Vital Delmas Mabonzo ◽  
Bienaime Bervi Bamvi Madzou ◽  
Melchior Josièrne Jupy Kokolo
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Simulations ◽  
Difference Method ◽  
Numerical Solutions ◽  
Classical Method ◽  
Elliptic Boundary ◽  
Boundary Condi Tions ◽  
Elliptic Boundary Value ◽  
Uniqueness Of The Solution

This paper refers to mathematical modelling and numerical analysis. The analysis to be presented through this paper deals with Robin’s problem which boundary equation is a linear combination of Dirichlet and Neumann-type boundary condi-tions. For this purpose we proved the existence and uniqueness of the solution. It is worth noting that the implementation of numerical simulations depends on the type of problem since it requires a search for explicit solution. Consequently, the motivation exists in this paper for choosing a classical method of variation of constants and employing a finite difference method to find the exact and numerical solutions, respectively so that numerical simulations were implemented in Scilab.

scholarly journals Mathematical Modelling and Numerical Simulation of Diffusive Processes in Slow Changing Domains

2020 ◽  
Author(s):  
Dmytro V. Yevdokymov ◽  
Yuri L. Menshikov
Keyword(s):  
Heat Conduction ◽  
Time Scale ◽  
Numerical Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Numerical Approach ◽  
Elliptic Boundary Value ◽  
Initial Boundary Value

Nowadays, diffusion and heat conduction processes in slow changing domains attract great attention. Slow-phase transitions and growth of biological structures can be considered as examples of such processes. The main difficulty in numerical solutions of correspondent problems is connected with the presence of two time scales. The first one is time scale describing diffusion or heat conduction. The second time scale is connected with the mentioned slow domain evolution. If there is sufficient difference in order of the listed time scale, strong computational difficulties in application of time-stepping algorithms are observed. To overcome the mentioned difficulties, it is proposed to apply a small parameter method for obtaining a new mathematical model, in which the starting parabolic initial-boundary-value problem is replaced by a sequence of elliptic boundary-value problems. Application of the boundary element method for numerical solution of the obtained sequence of problems gives an opportunity to solve the whole considered problem in slow time with high accuracy specific to the mentioned algorithm. Besides that, questions about convergence of the obtained asymptotic expansion and correspondence between initial and obtained formulations of the problem are considered separately. The proposed numerical approach is illustrated by several examples of numerical calculations for relevant problems.

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