NUMERICAL METHODS FOR DIFFUSION EQUATION ON DISTORTED POLYHEDRAL MESHES

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NUMERICAL METHODS FORDIFFUSION EQUATION
ONDISTORTED POLYHEDRAL MESHES
Yuri Kuznetsov Department of Mathematics Universi
ty of Houston http//lacsi.rice.edu/review/2004/sl
ides/Kuznetzov_LACSI_review.ppt
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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

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Outline

• Problem Formulation
• New Polyhedral Mesh Discretization
• Parallel Algebraic Solvers
• Applications and Numerical Results

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Diffusion Equation
Here
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First Order Differential Equations
p pressure, density, intensity, or
temperature u flux vector function
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Polyhedral Meshes
- polyhedral mesh cell
- boundary of
- interface between
and
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Polyhedral Mesh Cell E

• Gs - plane faces
• n - outward normal unit vector

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Examples of E

• Pyramids

Distored prisms
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Examples of E

• Distored cubes

Macro-Cell
cluster of distorted prisms
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Non-Matching Meshes

• Conformal Polyhedral Mortar Element
• Discretization VS Method

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Adaptive Mesh Refinement
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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

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Conservation Law
Discrete conservation law
where
Remark The discrete equation is exact
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Major Problem
to approximate the equation
or the equivalent variational equation
where v is a test vector function
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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

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Raviart-Thomas MFE Method on Tetrahedral Meshes
Basis vector functions
- affine vector-function
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Raviart-Thomas MFE Method on Tetrahedral Meshes
Then by
we get the discrete equations
where
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Genuine new MFE-method on arbitrary polyhedral
meshes
Let
be a partitioning of E into tetrahedrons
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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.
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Polyhedral Mimetic Finite Difference (MFD) Method
Kuznetsov-Lipnikov-Shashkov 2003-2004

• 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

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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
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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 Convergence rate 1.983 1.595
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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

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Algebraic Problem
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Condensed System
Here
-SPD matrix
where
-cell based matrices
-assembling matrices
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Algebraic Preconditioner

• Based on multilevel coarsening

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DISTORTED 2D-MESH
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Preconditioned Conjugate Gradient Method

• Stopping criterium

of iterations
Distortion factor q0.4
Distortion factor q0.3
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 q0.48
128x128 256x256 512x512
Diagonal Preconditioner 295 558 1106
Multilevel Preconditioner 21 27 30
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Distorted 3D Mesh
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Preconditioned Conjugate Gradient Method

• Stopping criterium

of iterations
Distortion factor q0.0
Distortion factor q0.2
16x16x16 32x32x32
Diagonal Preconditioner 78 153
Multilevel Preconditioner 27 28
16x16x16 32x32x32
Diagonal Preconditioner 108 205
Multilevel Preconditioner 28 29
Distortion factor q0.3
16x16x16 32x32x32
Diagonal Preconditioner 130 223
Multilevel Preconditioner 29 30
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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.

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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)

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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
• (Yu. Kuznetsov, J. Morel, M. Shashkov)

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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

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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. 265--284.
• 2. Yu. Kuznetsov and K. Lipnikov, Fast separable
  solvers for mixed finite element methods and
  applications, J. Numer. Math., 10 (2002), pp.
  137--155.
• 3. Yu. Kuznetsov, Spectrally equivalent
  preconditioners for mixed hybrid discretizations
  of diffusion equations on distorted meshes, J.
  Numer. Math., 11 (2003), pp. 61--74.
• 4. Yu. Kuznetsov and S. Repin, New mixed finite
  element method on polygonal and polyhedral
  meshes, RJNAMM, 18 (2003), pp. 261--278.
• 5. Yu. Kuznetsov and S. Repin, Mixed finite
  element methods on polygonal and polyhedral
  meshes, Proc. of the 5th ENUMATH Conference,
  Prague, 2003. 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).

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Current and Further Research

• Adaptive Refinement for Polyhedral Meshes
• Parallel Algebraic Solvers
• Polyhedral Discretizations for Radiation
  Transport Equations
• Maxwell Equations on Polyhedral Meshes