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Journal ArticleDOI

Precise FEM solution of a corner singularity using an adjusted mesh

10 Apr 2005-International Journal for Numerical Methods in Fluids (Wiley)-Vol. 47, pp 1285-1292

AbstractWithin the framework of the finite element method problems with corner-like singularities (e.g. on the well-known L-shaped domain) are most often solved by the adaptive strategy: the mesh near the corners is refined according to the a posteriori error estimates. In this paper we present an alternative approach. For flow problems on domains with corner singularities we use the a priori error estimates and asymptotic expansion of the solution to derive an algorithm for refining the mesh near the corner. It gives very precise solution in a cheap way. We present some numerical results. Copyright © 2005 John Wiley & Sons, Ltd.

Summary (1 min read)

Introduction

  • The local behaviour of the solution near the singular point is used to design a mesh which is adjusted to the shape of the solution.
  • Then the authors use this adjusted mesh for the numerical solution of flow in the channel with corners.

Model problem

  • The authors consider two-dimensional flow of viscous, incompressible fluid described by NavierStokes equations in a domain with corner singularity, cf. Fig.
  • Due to symmetry, the authors solve the problem only on the upper half of the channel.

Algorithm derivation

  • Similar results have been proved for the Navier-Stokes equations.
  • Using this algorithm the authors obtained satisfactory results.

Evaluation of the error

  • Here n means the number of all elements in the domain.
  • More about the evaluation of error in [2].

Numerical results

  • On Figures 5-8 the authors present the graphical output of entities that chartacterize the flow in the channel.
  • Figure 6 with contours of velocity vy shows that the solution is satisfactory smooth on refined area.
  • On Figs. 7-8 the authors observe how strong the singularity is, both for velocity and pressure (note that here the flow is from the right to the left, to have better view).
  • Fig. 6: Isolines of velocity vy Fig. 7: Velocity component vy Fig. 8: Pressure near the corner On Figs. 9-11 the authors show the errors on elements.

Conclusions

  • Presented results give satisfactory confirmation of developed algorithm.
  • Application of a priori error estimates of finite element method for mesh refinement near singularity is very efficient for their problem what can be seen especially from obtained errors on elements which are very uniformly distributed.
  • This approach is an alternative to ’classical’ one using adaptive mesh refinement, which is still much more robust.

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Citations
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Journal ArticleDOI
TL;DR: The results of computations of the lid-driven cavity problem show that the proposed method computes the central eddy with accuracy comparable to the best of existing methods and is more accurate for computing the corner eddies than the existing methods.
Abstract: It is well known that any viscous fluid flow near a corner consists of infinite series of eddies with decreasing size and intensity, unless the angle is larger than a certain critical angle [H. K. Moffat, J. Fluid Mech., 18 (1964), pp. 1-18]. The objective of the current work is to simulate such infinite series of eddies occurring in steady flows in domains with corners. The problem is approached by high-order finite element method with exponential mesh refinement near the corners, coupled with analytical asymptotics of the flow near the corners. Such approach allows one to compute position and intensity of the eddies near the corners in addition to the other main features of the flow. The method was tested on the problem of the lid-driven cavity flow as well as on the problem of the backward-facing step flow. The results of computations of the lid-driven cavity problem show that the proposed method computes the central eddy with accuracy comparable to the best of existing methods and is more accurate for computing the corner eddies than the existing methods. The results also indicate that the relative error of finding the eddies' intensity and position decreases uniformly for all the eddies as the mesh is refined (i.e., the relative error in computation of different eddies does not depend on their size).

21 citations


Journal ArticleDOI
TL;DR: The weighted analogue of the Ladyzhenskaya–Babuska–Brezzi condition is proved and a new weighted finite element method is constructed that shows the efficiency of the method.
Abstract: In this paper we introduce the notion of R ν -generalized solution to the Stokes problem with singularity in a non-convex polygonal domain with one reentrant corner of 3 π 2 on its boundary. The weighted analogue of the Ladyzhenskaya–Babuska–Brezzi condition is proved. A new weighted finite element method is constructed. Results of numerical experiments have shown the efficiency of the method.

18 citations


Journal ArticleDOI
Abstract: We deal with 2D flows of incompressible viscous fluids with high Reynolds numbers. Galerkin Least Squares technique of stabilization of the finite element method is studied and its modification is described. We present a number of numerical results obtained by the developed method, showing its contribution to solving flows with high Reynolds numbers. Several recommendations and remarks are included. We are interested in positive as well as negative aspects of stabilization, which cannot be divorced.

9 citations


Journal ArticleDOI
Abstract: In this study, a highly-accurate, conforming finite element method is developed and justified for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. The k − t h order Lagrange elements are used for the discretization of the variational form of the problem on exponentially compressed polar meshes employed in the neighbourhood of the corners whose interior angle is α π , α ≠ 1 ∕ 2 , and on the triangular and curved mesh formed in the remainder of the polygon. The exponentially compressed polar meshes are constructed such that they are transformed to square meshes using the Log-Polar transformation, simplifying the realization of the method significantly. For the error bound between the exact and the approximate solution obtained by the proposed method, an accuracy of O ( h k ) , h mesh size and k ≥ 1 an integer, is obtained in the H 1 -norm. Numerical experiments are conducted to support the theoretical analysis made. The proposed method can be applied for dealing with the corner singularities of general nonlinear parabolic partial differential equations with semi-implicit time discretization.

3 citations


01 Jan 2010
TL;DR: A parallel implementation of the Balancing Domain Decomposition by Constraints (BDDC) method is described that is based on formulation of BDDC with global matrices without explicit coarse problem and successfully applied to several problems of Stokes flow.
Abstract: The Balancing Domain Decomposition by Constraints (BDDC) method is one of the most advanced preconditioners for very large problems with symmetric positive definite matrices that arise from FEM analysis. It is very suitable for massively parallel computers. In presented paper, a parallel implementation of the method is described that is based on formulation of BDDC with global matrices without explicit coarse problem. It is based on the MUMPS parallel solver. The implementation is successfully applied to several problems of Stokes flow and BDDC is shown to be a very promising method also for this class of problems.

3 citations


Cites methods from "Precise FEM solution of a corner si..."

  • ...Flow in this geometry described by the Navier-Stokes model was studied in [3], with respect to precise solution of corner singularities....

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References
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Book
19 Jun 1986
TL;DR: This paper presents the results of an analysis of the "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions and its applications to Mixed Approximation and Homogeneous Stokes Equations.
Abstract: I. Mathematical Foundation of the Stokes Problem.- 1. Generalities on Some Elliptic Boundary Value Problems.- 1.1. Basic Concepts on Sobolev Spaces.- 1.2. Abstract Elliptic Theory.- 1.3. Example 1: Dirichlet's Problem for the Laplace Operator.- 1.4. Example 2: Neumann's Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- s Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- 2. Theory of the Steady-State Navier-Stokes Equations.- 2.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 2.2. The Stream Function Formulation of the Homogeneous Problem..- 3. Approximation of Branches of Nonsingular Solutions.- 3.1. An Abstract Framework.- 3.2. Approximation of Branches of Nonsingular Solutions.- 3.3. Application to a Class of Nonlinear Problems.- 3.4. Non-Differentiable Approximation of Branches of Nonsingular Solutions.- 4. Numerical Analysis of Centered Finite Element Schemes.- 4.1. Formulation in Primitive Variables: Methods Using Discontinuous Pressures.- 4.2. Formulation in Primitive Variables: the Case of Continuous Pressures.- 4.3. Mixed Incompressible Methods: the "Stream Function-Vorticity" Formulation.- 4.4. Remarks on the "Stream Function-Gradient of Velocity Tensor" Scheme.- 5. Numerical Analysis of Upwind Schemes.- 5.1. Upwinding in the Stream Function-Vorticity Scheme.- 5.2. Error Analysis of the Upwind Scheme.- 5.3. Approximating the Pressure with the Upwind Scheme.- 6. Numerical Algorithms.- 2.11. General Methods of Descent and Application to Gradient Methods.- 2.12. Least-Squares and Gradient Methods to Solve the Navier-Stokes Equations.- 2.13. Newton's Method and the Continuation Method.- References.- Index of Mathematical Symbols.

5,336 citations



Journal ArticleDOI
TL;DR: A posteriori error estimates for the Stokes and Navier-Stokes equations on two-dimensional polygonal domains are investigated and an incompressible flow problem in a domain with corners that cause singularities in the solution is solved.
Abstract: The paper consists of three parts. In the first part, we investigate a posteriori error estimates for the Stokes and Navier-Stokes equations on two-dimensional polygonal domains. Special attention is paid to the sources of the constants in the estimates, as these play a crucial role in practical applications to adaptive refinements, as we also show. In the second part, we deal with the problem of determining accurately the constants that appear in the estimates. We present a technique for calculating the constant with high accuracy. In the third part, we apply the a posteriori error estimates with the constants found numerically to the technique of adaptive mesh refinement-we solve an incompressible flow problem in a domain with corners that cause singularities in the solution.

16 citations


Frequently Asked Questions (1)
Q1. What are the contributions in "Precise fem solution of corner singularity using adjusted mesh applied to 2d flow" ?

The authors present an alternative approach to the adaptive mesh refinement. For steady Navier-Stokes equations the authors proved in [ 1 ] that for nonconvex internal angles the velocities near the corners possess an expansion u ( ρ, θ ) = ρφ ( θ ) +... ( + smoother terms ), where ρ, θ are local spherical coordinates. The authors show an example of 2D mesh with quadratic polynomials for velocity.