The $W^1_p$ stability of the Ritz projection on graded meshes

2016 ◽  
Vol 86 (303) ◽  
pp. 49-74 ◽  
Author(s):  
Hengguang Li
Keyword(s):  
Graded Meshes ◽  
Ritz Projection

scholarly journals Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

2020 ◽  
Vol 22 (3) ◽  
pp. 319-332
Author(s):  
Aleksandr N. Tynda ◽  
Konstantin A. Timoshenkov
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Spline Approximation ◽  
Boundary Integral ◽  
Double Layers ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Graded Meshes

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

Numerical Computations of Higher Class Solutions for 2D Polymer Flows Using PTT Constitutive Model

2001 ◽  
Author(s):  
K. S. Surana ◽  
H. Vijayendra Nayak
Keyword(s):  
Constitutive Model ◽  
Numerical Solutions ◽  
Research Work ◽  
Detailed Examination ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Stick Slip ◽  
Conservation Of Mass ◽  
Graded Meshes ◽  
Linear Algebraic Equations

Abstract This paper presents formulations, computations and investigations of the solutions of classes C00 and C11 for two dimensional viscoelastic fluid flows in u, v, p, τijp, τijs with Phan-Thien-Tanner (PTT) constitutive model using p-version least squares finite element formulation (LSFEF). The main thrust of the research work presented in the paper is to employ ‘right classes of interpolations’ and the ‘best computational strategy’ 1) to obtain numerical solutions of governing differential equations (GDEs) for increasing Deborah numbers 2) investigate the nature of the computed solutions with the aim of establishing limiting values of the flow parameters beyond which the solutions may be possible to compute, but may not be meaningful. The investigations presented in this paper reveal the following: a) The manner in which the stresses are non-dimensionalized significantly influences the performance of the iterative procedure of solving non-linear algebraic equations. b) Solutions of the class C00 are always the wrong class of solutions of GDEs in variables u, v, p, τijp and τijs and thus spurious. c) C11 class of solutions are the right class of solutions of the GDEs in variables u, v, p, τijp and τijs. d) In the flow domains, containing sharp gradients of the dependent variables, conservation of mass is difficult to achieve at lower p-levels (worse for coarse meshes). e) An augmented form of GDEs are proposed that always ensure conservation of mass at all p-levels regardless of the mesh and the nature of the solution gradients. f) Stick-slip problem is used as a model problem. We demonstrate that converged solutions are possible to compute for all flow rates reported and that the detailed examination of the solution characteristics reveals them to be in agreement with all the physics of the flow, g) Numerical studies with graded meshes and high p-levels presented in this paper are aimed towards establishing and demonstrating detail behavior of local as well as global nature of the computed solutions, h) Various norms are proposed and tested to judge local and global dominance of elasticity or viscous behavior i) New definitions are proposed for elongational (extensional) viscosity. The proposed definitions are more in conformity and agreement with the flow physics compared to currently used definitions j) A significant aspect and strength of our work is that we utilize straightforward p-version LSFEF with C00 and C11 type interpolations without linearizing GDEs and that SUPG, SUPG/DC, SUPG/DC/LS operators are neither needed nor used.

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