1. Department of Mathematics
NUMERICAL METHODS FOR DIFFUSION EQUATION ON DISTORTED POLYHEDRAL MESHES
Yuri Kuznetsov
Department of Mathematics
University of Houston

2. LACSI PROJECT: 2000-2004 PARALLEL NUMERICAL METHODS
FOR DIFFUSION EQUATIONS
IN HETEROGENEOUS MEDIA
ON STRONGLY DISTORED MESHES
LANL: J. Morel, K. Lipnikov, M. Shashkov
UH: Yu. Kuznetsov – PI
O. Boyarkin, V. Gvozdev, D. Svyatskiy – graduate students
S. Repin – visiting research professor

3. Outline Problem Formulation New Polyhedral Mesh Discretization
Parallel Algebraic Solvers
Applications and Numerical Results

4. Diffusion Equation
Here

5. First Order Differential Equations
p – pressure, density, intensity, or temperature
u – flux vector function

6. Polyhedral Meshes - polyhedral mesh cell - boundary of
- interface between
and

7. Gs - plane faces n - outward normal unit vector
Polyhedral Mesh Cell E
Gs - plane faces
n - outward normal unit vector

8. Examples of E
Pyramids
Distored prisms

9. Examples of E
Distored cubes
Macro-Cell
cluster of distorted prisms

10. Non-Matching Meshes Conformal Polyhedral Mortar Element
Discretization VS Method

11. Adaptive Mesh Refinement

12. Requirements For Discretization
arbitrary polyhedral cells including nonconvex and degenerated ones
arbitrary symmetric positive definite diffusion tensor
one Degree of Freedom (DOF) per cell for the solution function
one DOF per interface for the solution function
one DOF per interface for the flux vector-function

13. Discrete conservation law
where
Remark: The discrete equation is exact

14. Major Problem to approximate the equation
or the equivalent variational equation
where v is a test vector function

15. Drawbacks of Existing Polyhedral Mesh Discretizations
Mixed Finite Element (MFE) Method
only tetrahedral meshes (classical variant)
only distorted convex prismatic and cubic cells (variant with Piola Transformation)
Finite Volume (FV) Method
only convex polyhedrons
very low accuracy on strongly distorted polyhedrons
low accuracy for nonscalar diffusion tensor
results in nonsymmetric matrices

16. Raviart-Thomas MFE Method on Tetrahedral Meshes
Basis vector functions
- affine vector-function

17. Raviart-Thomas MFE Method on Tetrahedral Meshes
Then by
we get the discrete equations
where

18. Genuine new MFE-method on arbitrary polyhedral meshes
Let
be a partitioning of E into tetrahedrons

19. Kuznetsov-Repin, 2003 To design the basis vector-functions
we solve the local diffusion problem
with the Neumann-type boundary conditions
by the Raviart-Thomas MFE method on the tetrahedral mesh
Here,
is the positive constant.

20. Polyhedral Mimetic Finite Difference (MFD) Method
Kuznetsov-Lipnikov-Shashkov:
MFD-method mimics the most important properties of underlying physical and mechanical models, e.g., conservation laws, as well as geometrical and mathematical symmetries.
earlier versions did not allow nonconvex and degenerated polyhedrons

21. Numerical Experiments
1/h ep ef 4
2.647e-2
1.330e-1 8
9.704e-3
4.587e-2 16
2.671e-3
1.233e-2 32
6.860e-4
3.155e-3
Convergence
rate
1.767
1.808

22. Numerical Experiments
1/h1
1/h2 ep ef 7 5
1.604e-4
3.502e-3 14 10
4.078e-5
1.144e-3 28 20
1.025e-5
3.837e-4
Convergence rate
1.983
1.595

23. Advantages of the new method
Arbitrary polyhedral meshes including meshes with nonconvex and degenerating cells
Non-matching polyhedral meshes including AMR ones
Arbitrary diffusion tensor
Major restriction
For accuracy reason the interface boundaries Gkl between different polyhedral cells Ek and El should be plane or "almost plane" polygons

24. Algebraic Problem

25. Condensed System Here -SPD matrix where -cell based matrices
-assembling matrices

26. Algebraic Preconditioner
Based on multilevel coarsening

27. DISTORTED 2D-MESH

28. Preconditioned Conjugate Gradient Method
Stopping criterium:
# of iterations
Distortion factor q=0.4
Distortion factor q=0.3
128x128
256x256
512x512
Diagonal Preconditioner
227
455
882
Multilevel Preconditioner 21 25 29
128x128
256x256
512x512
Diagonal Preconditioner
262
483
967
Multilevel Preconditioner 21 26 30
Distortion factor q=0.48
128x128
256x256
512x512
Diagonal Preconditioner
295
558
1106
Multilevel Preconditioner 21 27 30

29. Distorted 3D Mesh

30. Preconditioned Conjugate Gradient Method
Stopping criterium:
# of iterations
Distortion factor q=0.0
Distortion factor q=0.2
16x16x16
32x32x32
Diagonal Preconditioner 78
153
Multilevel Preconditioner 27 28
16x16x16
32x32x32
Diagonal Preconditioner
108
205
Multilevel Preconditioner 28 29
Distortion factor q=0.3
16x16x16
32x32x32
Diagonal Preconditioner
130
223
Multilevel Preconditioner 29 30

31. ASC Relevant Application
LANL researchers M. Shashkov and K. Lipnikov are now working with an X-3 team (S. Runnels) to implement the 3D geometry version of the new polyhedral mesh discretization to model diffusion processes in Project B.
The scheme currently existing is nonsymmetric.
It is expected that the new scheme will be much more accurate, and it will cost far less to solve the underlying symmetric positive definite algebraic systems by Preconditioned Conjugate Gradient (PCG) method than the restarted GMRES method.

32. Other Applications
Basin Modeling and Heat Transport in Strongly Heterogeneous Media on Highly Distorted Polyhedral Meshes (ExxonMobil Upstream Research Co.)
Numerical Simulation for Nuclear Waste Deposit (INRIA, France)

33. Joint Workshops LACSI Symposium 2003
Mimetic Methods for Radiation Transport and Diffusion
(Yu. Kuznetsov, J. Morel, M. Shashkov)
SIAM Conference on Mathematical and Computational Issues in Geosciences
(Austin, 2003)
Discretizations and Iterative Solvers for Diffusion Problems in Strongly Heterogeneous Media
(Yu. Kuznetsov, M. Shashkov)
LACSI Symposium 2004
Mimetic Methods for Partial Differential Equations and Applications

34. EDUCATION ISSUES UH students on summer semesters at LANL
2001: K. Lipnikov, A. Hayrapetyan
2002: K. Lipnikov, V. Dyadechko
2003: V. Dyadechko
2004: D. Svyatskiy
Ph.D. Thesis
2002: K. Lipnikov - currently a PostDoc at T7, LANL; considered for a tenure research position at LANL
2003: V. Dyadechko - currently a PostDoc at T7, LANL

35. List of Publications
1. M. Berndt, K. Lipnikov, D. Moulton, and M. Shashkov, Convergence of mimetic difference discretizations of the diffusion equations, J. Numer. Math., 9 (2001), pp
2. Yu. Kuznetsov and K. Lipnikov, Fast separable solvers for mixed finite element methods and applications, J. Numer. Math., 10 (2002), pp
3. Yu. Kuznetsov, Spectrally equivalent preconditioners for mixed hybrid discretizations of diffusion equations on distorted meshes, J. Numer. Math., 11 (2003), pp
4. Yu. Kuznetsov and S. Repin, New mixed finite element method on polygonal and polyhedral meshes, RJNAMM, 18 (2003), pp
5. Yu. Kuznetsov and S. Repin, Mixed finite element methods on polygonal and polyhedral meshes, Proc. of the 5th ENUMATH Conference, Prague, World Scientific Publ. Co., 2004.
6. Yu. Kuznetsov, K. Lipnikov, and M. Shashkov, Mimetic finite difference equations on polygonal meshes for diffusion-type equations, Comput. Geosciences, 6 (2005).

36. Current and Further Research
Adaptive Refinement for Polyhedral Meshes
Parallel Algebraic Solvers
Polyhedral Discretizations for Radiation Transport Equations
Maxwell Equations on Polyhedral Meshes