tomás chacón rebollo macarena gómez mármol gladys narbona reina university of seville
DESCRIPTION
Numerical analysis of PSI method via Petrov-Galerkin formulation: Application to convection-diffusion and Navier Stokes equations. Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville. European Finite Element Fair 2007. Motivation. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/1.jpg)
Numerical analysis of PSI Numerical analysis of PSI method via Petrov-Galerkin method via Petrov-Galerkin formulation: Application to formulation: Application to
convection-diffusion and Navier convection-diffusion and Navier Stokes equations.Stokes equations.
Tomás Chacón RebolloTomás Chacón RebolloMacarena Gómez MármolMacarena Gómez Mármol
Gladys Narbona ReinaGladys Narbona Reina
University of SevilleUniversity of SevilleEuropean European
Finite Finite Element Element Fair 2007Fair 2007
![Page 2: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/2.jpg)
MotivationMotivation
PSI and in general Fluctuation Splitting (FS) methods provide efficient numerical solutions of flow problems: Accurate and Positive.
First order (linear) FS methods: Cast and analyzed as Finite Volume methods by Perthame and co-workers (1994, 1995).
Here we cast and analyze “second” order (non-linear) FS method as Petrov-Galerkin methods.
![Page 3: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/3.jpg)
(P)
Continuous problem:
Convection-diffusion problemConvection-diffusion problem
![Page 4: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/4.jpg)
Discrete problemDiscrete problem:
Convection-Diffusion Convection-Diffusion problemproblem
Discrete space for convection test functions
Distributed Interpolation operator:
![Page 5: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/5.jpg)
bi
Convection-Diffusion Convection-Diffusion problemproblem
With:With:
(Ph)
We consider two possible discretizations:
To obtain a well-balanced scheme for conv. domin. regime
![Page 6: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/6.jpg)
Hypothesis 1:
Hypothesis 2:
HypothesesHypotheses
Hypothesis 3:
![Page 7: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/7.jpg)
1 downstream node
2 downstream nodes
Fluctuation to node bi
Treatment of convective term:
Idea: send the convective flow to downstream nodes:
Relationship with Fluctuation Splitting Relationship with Fluctuation Splitting methodsmethods
![Page 8: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/8.jpg)
We define the associated piecewise constant basis function i by:
FS methodsFS methods
Thus
![Page 9: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/9.jpg)
Error estimates for functions:
Stability:
Analysis: Analysis: Properties of the Distribution Properties of the Distribution Interpolation OperatorInterpolation Operator
Under Hypotheses 1 and 2:
![Page 10: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/10.jpg)
Positiveness of the discrete convection operatorPositiveness of the discrete convection operator
The form
is bilinear, bounded:
and semi-positive:
![Page 11: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/11.jpg)
Existence: Problem (Ph) admits at least one solution that satifies:
This solution is quasi-unique: If 1h and 2h are two solutions of (Ph), then :
Theoretical results for convection-diffusion Theoretical results for convection-diffusion problemproblem
![Page 12: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/12.jpg)
Error estimates:
Convergence: The sequence of solutions {h}h of (Ph), is strongly convergent in V to the solution of problem (P).
Convergence AnalysisConvergence Analysis
In the best case, this error is O(h). This occurs when u is in L1 and f is in L2.
![Page 13: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/13.jpg)
Well-balance property: Consider rh2 Vh some interpolate of , and the consistency error given by
Maximum principle: Under the hypotheses above, assume f ¸ 0. Then h ¸ 0.
Other properties of interestOther properties of interest
Then, . Or, in other words, the upwinded scheme is well-balanced up to second order for convection-dominated regimes.
![Page 14: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/14.jpg)
Continuous problem:
Navier-Stokes equationsNavier-Stokes equations
We denote by the norm of the form:
![Page 15: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/15.jpg)
Discrete problem:
Navier-Stokes equationsNavier-Stokes equations
![Page 16: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/16.jpg)
The extension The extension of Distributed Interpolation operator is made by components is made by components, so we define::
And denote by the vectorialvectorial Distributed Interpolation operator:
Thus, using to test the convection term:
![Page 17: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/17.jpg)
Hypothesis 2’: For each component, the matrix of convective term, C is semi-positive defined. It is given by:
HypothesesHypotheses
If rh belongs to Vh, the discrete convection
ToolsTools
is semi-positive:
![Page 18: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/18.jpg)
Theoretical results for Navier-Stokes equationsTheoretical results for Navier-Stokes equations
where
Existence: The discrete problem admits at least one solution that satisfies:
![Page 19: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/19.jpg)
Convergence: There exists a subsequence of {(uh,ph)}h solution of (Ph) that converges strongly in V £ L0
2() to a solution of (P).
Navier-Stokes equationsNavier-Stokes equations
Error estimates: We suppose that:
Then there exist a positive constant C such that:
![Page 20: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/20.jpg)
Numerical resultsNumerical resultsTest 1: Test 1: CavityCavity
Data: =[0,1]2, Re=1000, f=0. Initial.C.: u=0. Boundary C.:
Characteristic method
PSI method
![Page 21: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/21.jpg)
Numerical resultsNumerical results
Test 2: Forward StepTest 2: Forward Step
Data: Re=400, f=0. Initial.C.: u=0. Boundary C.:
Characteristic method
PSI method
4
3
2
1
![Page 22: Tomás Chacón Rebollo Macarena Gómez Mármol Gladys Narbona Reina University of Seville](https://reader035.vdocument.in/reader035/viewer/2022070414/56814dca550346895dbb203a/html5/thumbnails/22.jpg)
Numerical resultsNumerical results
Test 3: ObstacleTest 3: Obstacle
Data: Re=1000, f=0. I.C.: . B. C.:
Characteristic method
PSI method
4
3
2
1