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March  2021, 14(3): 865-879. doi: 10.3934/dcdss.2020350

## Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh

 1 Faculty of Civil Engineering, Slovak University of Technology, Department of Mathematics and Descriptive Geometry, Radlinského 11,810 05 Bratislava, Slovak Republic 2 Advanced Simulation Technologies, AVL List GmbH, Hans-List Platz 1, 8010 Graz, Austria

* Corresponding author: Peter Frolkovič

Received  January 2019 Revised  November 2019 Published  May 2020

Fund Project: The first and second author are supported by grants VEGA 1/0709/19 and 1/0436/20 and APVV-15-0522

A numerical method for solving diffusion problems on polyhedral meshes is presented. It is based on a finite volume approximation with the degrees of freedom located in the centers of computational cells. A numerical gradient is defined by a least-squares minimization for each cell, where we suggest a restricted form in the case of discontinuous diffusion coefficient. The flux balanced approximation is proposed without numerically computing the gradient itself at the faces of computational cells in order to find a normal diffusive flux. To apply the method for parallel computations with a 1-ring neighborhood, we use an iterative method to solve the obtained system of algebraic equations. Several numerical examples illustrate some advantages of the proposed method.

Citation: Peter Frolkovič, Karol Mikula, Jooyoung Hahn, Dirk Martin, Branislav Basara. Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 865-879. doi: 10.3934/dcdss.2020350
##### References:
 [1] L. Alvarez, P.-L. Lions and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. Ⅱ, SIAM J. Numer. Anal., 29 (1992), 845-866.  doi: 10.1137/0729052.  Google Scholar [2] M. Balažovjech and K. Mikula, A higher order scheme for a tangentially stabilized plane curve shortening flow with a driving force, SIAM J. Sci. Comput., 33 (2011), 2277-2294.  doi: 10.1137/100795309.  Google Scholar [3] B. Basara, Employment of the second-moment turbulence closure on arbitrary unstructured grids, Int. J. Numer. Methods Fluids, 44 (2004), 377-407.  doi: 10.1002/fld.646.  Google Scholar [4] P. Bastian, Numerical Computation of Multiphase Flows in Porous Media, Ph.D thesis, Habilitationsschrift Univeristät Kiel, 1999. Google Scholar [5] J. Blazek, Computational Fluid Dynamics: Principles and Applications, Elsevier/Butterworth Heinemann, Amsterdam, 2015.  Google Scholar [6] K. Böhmer, P. Hemker and H. J. 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Street, Computational Methods for Fluid Dynamics, Springer, Cham, 2020. doi: 10.1007/978-3-319-99693-6.  Google Scholar [15] A. Fluent, Release 15.0, Theory Guide, November. Google Scholar [16] P. Frolkovič, M. Lampe and G. Wittum, Numerical simulation of contaminant transport in groundwater using software tools of $r^3t$, Comput. Vis. Sci., 18 (2016), 17-29.  doi: 10.1007/s00791-016-0268-0.  Google Scholar [17] H. Jasak, Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, Ph.D thesis, Imperial College London (University of London), 1996. Google Scholar [18] H. Jasak and A. D. Gosman, Automatic resolution control for the finite volume method. Part 1, Numer. Heat Tr. B-Fund., 38 (2000), 237-256.  doi: 10.1080/10407790050192753.  Google Scholar [19] H. Jasak, A. Jemcov, Z. Tukovic et al., Openfoam: A C++ library for complex physics simulations, in International Workshop on Coupled Methods in Numerical Dynamics, 1000, IUC Dubrovnik, Croatia, 2007, 1–20. Google Scholar [20] J. Jin, The Finite Element Method in Electromagnetics, John Wiley & Sons, New York, 2012.  Google Scholar [21] J. Kačur and K. Mikula, Solution of nonlinear diffusion appearing in image smoothing and edge detection, Appl. Numer. Math., 17 (1995), 47-59.  doi: 10.1016/0168-9274(95)00008-I.  Google Scholar [22] S. R. Mathur and J. Y. Murthy, A pressure-based method for unstructured meshes, Numer. Heat Tr. B-Fund., 31 (1997), 195-215.  doi: 10.1080/10407799708915105.  Google Scholar [23] S. Muzaferija, Adaptive Finite Volume Method for Flow Prediction Using Unstructured Meshes and Multigrid Approach, Ph.D thesis, University of London, 1994. Google Scholar [24] S. Muzaferija and D. Gosman, Finite-volume CFD procedure and adaptive error control strategy for grids of arbitrary topology, J. Comput. Phys., 138 (1997), 766-787.  doi: 10.1006/jcph.1997.5853.  Google Scholar [25] B. Niceno, A three dimensional finite volume method for incompressible Navier-Stokes equations on unstructured staggered grids, ECCOMAS CFP, 2006. Google Scholar [26] E. Sozer, C. Brehm and C. C. Kiris, Gradient calculation methods on arbitrary polyhedral unstructured meshes for cell-centered CFD solvers, 52nd Aerospace Sciences Meeting, 2014. doi: 10.2514/6.2014-1440.  Google Scholar [27] R. Tatschl, B. Basara, J. Schneider, K. Hanjalic, M. Popovac, A. Brohmer and J. Mehring, Advanced turbulent heat transfer modeling for IC-engine applications using AVL FIRE, in Int. Multidimensional Engine Modelling, 2, User's Group Meeting, Detroit, MI, 2006, 1–10. Google Scholar [28] Y.-Y. Tsui and Y.-F. Pan, A pressure-correction method for incompressible flows using unstructured meshes, Numer. Heat Tr. B-Fund., 49 (2006), 43-65.  doi: 10.1080/10407790500344084.  Google Scholar [29] J. Tu, G.-H. Yeoh and C. Liu, Computational Fluid Dynamics: A Practical Approach, Elsevier/Butterworth Heinemann, Amsterdam, 2013. doi: 10.1016/B978-0-08-098243-4.00001-9.  Google Scholar [30] L. White, R. Panchadhara and D. Trenev, Flow simulation in heterogeneous porous media with the moving least-squares method, SIAM J. Sci. Comput., 39 (2017), B323–B351. doi: 10.1137/16M1070840.  Google Scholar

show all references

##### References:
 [1] L. Alvarez, P.-L. Lions and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. Ⅱ, SIAM J. Numer. Anal., 29 (1992), 845-866.  doi: 10.1137/0729052.  Google Scholar [2] M. Balažovjech and K. Mikula, A higher order scheme for a tangentially stabilized plane curve shortening flow with a driving force, SIAM J. Sci. Comput., 33 (2011), 2277-2294.  doi: 10.1137/100795309.  Google Scholar [3] B. Basara, Employment of the second-moment turbulence closure on arbitrary unstructured grids, Int. J. Numer. Methods Fluids, 44 (2004), 377-407.  doi: 10.1002/fld.646.  Google Scholar [4] P. Bastian, Numerical Computation of Multiphase Flows in Porous Media, Ph.D thesis, Habilitationsschrift Univeristät Kiel, 1999. Google Scholar [5] J. Blazek, Computational Fluid Dynamics: Principles and Applications, Elsevier/Butterworth Heinemann, Amsterdam, 2015.  Google Scholar [6] K. Böhmer, P. Hemker and H. J. Stetter, The defect correction approach, in Defect Correction Methods, Comput. Suppl., 5, Springer, Vienna, 1984, 1–32. doi: 10.1007/978-3-7091-7023-6_1.  Google Scholar [7] N. Cinosi, S. Walker, M. Bluck and R. Issa, CFD simulation of turbulent flow in a rod bundle with spacer grids (MATIS-H) using STAR-CCM+, Nuclear Engrg. Design, 279 (2014), 37-49.  doi: 10.1016/j.nucengdes.2014.06.019.  Google Scholar [8] A. de Boer, M. S. van der Schoot and H. Bijl, Mesh deformation based on radial basis function interpolation, Comput. Structures, 85 (2007), 784-795.  doi: 10.1016/j.compstruc.2007.01.013.  Google Scholar [9] I. Demirdžić, On the discretization of the diffusion term in finite-volume continuum mechanics, Numer. Heat Tr. B-Fund., 68 (2015), 1-10.  doi: 10.1080/10407790.2014.985992.  Google Scholar [10] I. Demirdžić, I. Horman and D. Martinović, Finite volume analysis of stress and deformation in hygro-thermo-elastic orthotropic body, Comp. Meth. Appl. Mech. Engrg., 190 (2000), 1221-1232.  doi: 10.1016/S0045-7825(99)00476-4.  Google Scholar [11] I. Demirdžić and S. Muzaferija, Numerical method for coupled fluid flow, heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology, Comp. Meth. Appl. Mech. Engrg., 125 (1995), 235-255.  doi: 10.1016/0045-7825(95)00800-G.  Google Scholar [12] J. Droniou, R. Eymard, T. Gallouët and R. Herbin, A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods, Math. Models Methods Appl. Sci., 20 (2010), 265-295.  doi: 10.1142/S0218202510004222.  Google Scholar [13] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, Vol. VII, Handb. Numer. Anal., Ⅶ, North-Holland, Amsterdam, 2000,713–1020. doi: 10.1086/phos.67.4.188705.  Google Scholar [14] J. H. Ferziger, M. Perić and R. L. Street, Computational Methods for Fluid Dynamics, Springer, Cham, 2020. doi: 10.1007/978-3-319-99693-6.  Google Scholar [15] A. Fluent, Release 15.0, Theory Guide, November. Google Scholar [16] P. Frolkovič, M. Lampe and G. Wittum, Numerical simulation of contaminant transport in groundwater using software tools of $r^3t$, Comput. Vis. Sci., 18 (2016), 17-29.  doi: 10.1007/s00791-016-0268-0.  Google Scholar [17] H. Jasak, Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, Ph.D thesis, Imperial College London (University of London), 1996. Google Scholar [18] H. Jasak and A. D. Gosman, Automatic resolution control for the finite volume method. Part 1, Numer. Heat Tr. B-Fund., 38 (2000), 237-256.  doi: 10.1080/10407790050192753.  Google Scholar [19] H. Jasak, A. Jemcov, Z. Tukovic et al., Openfoam: A C++ library for complex physics simulations, in International Workshop on Coupled Methods in Numerical Dynamics, 1000, IUC Dubrovnik, Croatia, 2007, 1–20. Google Scholar [20] J. Jin, The Finite Element Method in Electromagnetics, John Wiley & Sons, New York, 2012.  Google Scholar [21] J. Kačur and K. Mikula, Solution of nonlinear diffusion appearing in image smoothing and edge detection, Appl. Numer. Math., 17 (1995), 47-59.  doi: 10.1016/0168-9274(95)00008-I.  Google Scholar [22] S. R. Mathur and J. Y. Murthy, A pressure-based method for unstructured meshes, Numer. Heat Tr. B-Fund., 31 (1997), 195-215.  doi: 10.1080/10407799708915105.  Google Scholar [23] S. Muzaferija, Adaptive Finite Volume Method for Flow Prediction Using Unstructured Meshes and Multigrid Approach, Ph.D thesis, University of London, 1994. Google Scholar [24] S. Muzaferija and D. Gosman, Finite-volume CFD procedure and adaptive error control strategy for grids of arbitrary topology, J. Comput. Phys., 138 (1997), 766-787.  doi: 10.1006/jcph.1997.5853.  Google Scholar [25] B. Niceno, A three dimensional finite volume method for incompressible Navier-Stokes equations on unstructured staggered grids, ECCOMAS CFP, 2006. Google Scholar [26] E. Sozer, C. Brehm and C. C. Kiris, Gradient calculation methods on arbitrary polyhedral unstructured meshes for cell-centered CFD solvers, 52nd Aerospace Sciences Meeting, 2014. doi: 10.2514/6.2014-1440.  Google Scholar [27] R. Tatschl, B. Basara, J. Schneider, K. Hanjalic, M. Popovac, A. Brohmer and J. Mehring, Advanced turbulent heat transfer modeling for IC-engine applications using AVL FIRE, in Int. Multidimensional Engine Modelling, 2, User's Group Meeting, Detroit, MI, 2006, 1–10. Google Scholar [28] Y.-Y. Tsui and Y.-F. Pan, A pressure-correction method for incompressible flows using unstructured meshes, Numer. Heat Tr. B-Fund., 49 (2006), 43-65.  doi: 10.1080/10407790500344084.  Google Scholar [29] J. Tu, G.-H. Yeoh and C. Liu, Computational Fluid Dynamics: A Practical Approach, Elsevier/Butterworth Heinemann, Amsterdam, 2013. doi: 10.1016/B978-0-08-098243-4.00001-9.  Google Scholar [30] L. White, R. Panchadhara and D. Trenev, Flow simulation in heterogeneous porous media with the moving least-squares method, SIAM J. Sci. Comput., 39 (2017), B323–B351. doi: 10.1137/16M1070840.  Google Scholar
An illustration of the basic notation used in the flux approximation
The cubic domain and polyhedral mesh with average discretization size $\ell = .19$ used in the Sections 4.1 and 4.4
The domain used for the examples in the Section 4.2. The left picture shows a cut for which the mesh is visible. The blue and red parts visualize the subdomains with the different constant diffusion coefficients. The right picture shows the outer boundary with the corresponding surface mesh
The cut (left) and the surface (right) of the complex domain used for the example in the Section 4.3
The three surfaces of the boundary of computational domain (left) and the isosurfaces of numerical solution (right) for the example in the Section 4.3
The error norms and EOCs for the example in the Section 4.1
 $\ell$ $K$ $E_2$ EOC $G_1$ EOC 1.90e-1 10 3.69e-3 7.97e-2 9.52e-2 10 1.17e-3 1.66 3.22e-2 1.31 4.76e-2 9 3.27e-4 1.85 1.41e-2 1.19 2.48e-2 9 7.53e-5 2.26 6.08e-3 1.30
 $\ell$ $K$ $E_2$ EOC $G_1$ EOC 1.90e-1 10 3.69e-3 7.97e-2 9.52e-2 10 1.17e-3 1.66 3.22e-2 1.31 4.76e-2 9 3.27e-4 1.85 1.41e-2 1.19 2.48e-2 9 7.53e-5 2.26 6.08e-3 1.30
The error norms and the EOCs for the example in the concentric spherical domain in the Section 4.2 for the flux balanced method. The full least square gradient approximation is presented in the columns from $2$ to $6$, and the restricted one in the columns from $7$ to $11$
 $\ell$ $K$ $E_2$ EOC $G_1$ EOC $K$ $E_2$ EOC $G_1$ EOC .121 8 6.43e-5 3.41e-3 8 6.43e-5 1.07e-3 .090 8 3.08e-5 2.42 2.26e-3 1.36 8 3.08e-5 2.42 5.79e-4 2.03 .072 8 1.80e-5 2.44 1.70e-3 1.29 8 1.80e-5 2.44 3.93e-4 1.75 .060 8 1.15e-5 2.46 1.34e-3 1.27 8 1.15e-5 2.46 2.88e-4 1.71
 $\ell$ $K$ $E_2$ EOC $G_1$ EOC $K$ $E_2$ EOC $G_1$ EOC .121 8 6.43e-5 3.41e-3 8 6.43e-5 1.07e-3 .090 8 3.08e-5 2.42 2.26e-3 1.36 8 3.08e-5 2.42 5.79e-4 2.03 .072 8 1.80e-5 2.44 1.70e-3 1.29 8 1.80e-5 2.44 3.93e-4 1.75 .060 8 1.15e-5 2.46 1.34e-3 1.27 8 1.15e-5 2.46 2.88e-4 1.71
The error norms and the EOCs for the example in the concentric spherical domain in the Section 4.2 using the method based on the face gradient approximation. The full least square gradient approximation is presented in the columns from $2$ to $6$, and the restricted one in the columns from $7$ to $11$
 $\ell$ $K$ $E_2$ EOC $G_1$ EOC $K$ $E_2$ EOC $G_1$ EOC .121 10 2.99e-4 6.44e-3 10 2.83e-4 3.49e-3 .090 9 2.07e-4 1.21 4.53e-3 1.16 9 1.94e-4 1.23 2.49e-3 1.11 .072 9 1.59e-4 1.20 3.48e-3 1.18 9 1.48e-4 1.22 1.92e-3 1.17 .060 9 1.28e-4 1.16 2.81e-3 1.16 9 1.19e-4 1.19 1.56e-3 1.16
 $\ell$ $K$ $E_2$ EOC $G_1$ EOC $K$ $E_2$ EOC $G_1$ EOC .121 10 2.99e-4 6.44e-3 10 2.83e-4 3.49e-3 .090 9 2.07e-4 1.21 4.53e-3 1.16 9 1.94e-4 1.23 2.49e-3 1.11 .072 9 1.59e-4 1.20 3.48e-3 1.18 9 1.48e-4 1.22 1.92e-3 1.17 .060 9 1.28e-4 1.16 2.81e-3 1.16 9 1.19e-4 1.19 1.56e-3 1.16
The error norms and the EOCs for the example with Perona-Malik equation in the Section 4.4
 $\ell$ $K$ $E_2$ EOC $G_1$ EOC 1.90e-1 7 1.84e-4 1.70e-2 9.52e-2 6 5.31e-5 1.79 6.66e-3 1.36 4.76e-2 6 1.55e-5 1.78 2.96e-3 1.17 2.48e-2 5 4.03e-6 2.07 1.29e-3 1.28
 $\ell$ $K$ $E_2$ EOC $G_1$ EOC 1.90e-1 7 1.84e-4 1.70e-2 9.52e-2 6 5.31e-5 1.79 6.66e-3 1.36 4.76e-2 6 1.55e-5 1.78 2.96e-3 1.17 2.48e-2 5 4.03e-6 2.07 1.29e-3 1.28
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