Goal-Oriented Anisotropic hp-Adaptive Discontinuous Galerkin Method for the Euler Equations

doi:10.1007/s42967-020-00102-5

Abstract

Abstract: We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin (DG) method with focus on the goal-oriented error estimates and adaptivity. We analyse the adjoint consistency of the DG scheme where the adjoint problem is not formulated by the diferentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms. Furthermore, we present the goaloriented anisotropic hp-mesh adaptation method for the Euler equations. The theoretical results are supported by numerical experiments.

Key words: Euler equations, Discontinuous Galerkin method, Target functional, Adjoint consistency, Anisotropic hp-mesh adaptation

CLC Number:

Vít Dolejší, Filip Roskovec. Goal-Oriented Anisotropic hp-Adaptive Discontinuous Galerkin Method for the Euler Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 143-179.

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URL: https://www.camc.shu.edu.cn/EN/10.1007/s42967-020-00102-5

https://www.camc.shu.edu.cn/EN/Y2022/V4/I1/143

References

1. Balan, A., Woopen, M., May, G.:Adjoint-based hp-adaptivity on anisotropic meshes for high-order compressible fow simulations. Comput. Fluids 139, 47-67 (2016)
2. Bangerth, W., Rannacher, R.:Adaptive Finite Element Methods for Diferential Equations. Lectures in Mathematics. ETH Zürich, Birkhäuser Verlag, Zurich (2003)
3. Bartoš, O., Dolejší, V., May, G., Rangarajan, A., Roskovec, F.:Goal-oriented anisotropic hp-mesh optimization technique for linear convection-difusion-reaction problem. Comput. Math. Appl. 78(9), 2973-2993 (2019)
4. Bassi, F., Rebay, S.:High-order accurate discontinuous fnite element solution of the 2D Euler equations. J. Comput. Phys. 138, 251-285 (1997)
5. Bassi, F., Rebay, S.:Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 40, 197-207 (2002)
6. Becker, R., Rannacher, R.:An optimal control approach to a-posteriori error estimation in fnite element methods. Acta Numer. 10, 1-102 (2001)
7. Cao, W.:On the error of linear interpolation and the orientation, aspect ratio, and internal angles of a triangle. SIAM J. Numer. Anal. 43(1), 19-40 (2005)
8. Ceze, M., Fidkowski, K.J.:Anisotropic hp-adaptation framework for functional prediction. AIAA J. 51(2), 492-509 (2012)
9. Dolejší, V.:A design of residual error estimates for a high order BDF-DGFE method applied to compressible fows. Int. J. Numer. Methods Fluids 73(6), 523-559 (2013)
10. Dolejší, V.:Anisotropic hp-adaptive method based on interpolation error estimates in the L^q-norm. Appl. Numer. Math. 82, 80-114 (2014)
11. Dolejší, V., Feistauer, M.:Semi-implicit discontinuous Galerkin fnite element method for the numerical solution of inviscid compressible fow. J. Comput. Phys. 198(2), 727-746 (2004)
12. Dolejší, V., Feistauer, M.:Discontinuous Galerkin Method-Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics, vol. 48. Springer, Cham (2015)
13. Dolejší, V., May, G., Rangarajan, A., Roskovec, F.:A goal-oriented high-order anisotropic mesh adaptation using discontinuous Galerkin method for linear convection-difusion-reaction problems. SIAM J. Sci. Comput. 41(3), A1899-A1922 (2019)
14. Dolejší, V., Roskovec, F.:Goal-oriented error estimates including algebraic errors in discontinuous Galerkin discretizations of linear boundary value problems. Appl. Math. 62(6), 579-605 (2017)
15. Dolejší, V., Roskovec, F., Vlasák, M.:Residual based error estimates for the space-time discontinuous Galerkin method applied to the compressible fows. Comput. Fluids 117, 304-324 (2015)
16. Feistauer, M., Kučera, V.:On a robust discontinuous Galerkin technique for the solution of compressible fow. J. Comput. Phys. 224, 208-221 (2007)
17. Feistauer, M., Felcman, J., Straškraba, I.:Mathematical and Computational Methods for Compressible Flow. Clarendon Press, Oxford (2003)
18. Fidkowski, K., Darmofal, D.:Review of output-based error estimation and mesh adaptation in computational fuid dynamics. AIAA J. 49(4), 673-694 (2011)
19. Fidkowski, K.J., Luo, Y.:Output-based space-time mesh adaptation for the compressible Navier-Stokes equations. J. Comput. Phys. 230(14), 5753-5773 (2011)
20. Georgoulis, E.H., Hall E., Houston, P.:Discontinuous Galerkin methods for advection-difusion-reaction problems on anisotropically refned meshes. SIAM J. Sci. Comput. 30(1), 246-271 (2007)
21. Giani, S., Houston, P.:Anisotropic hp-adaptive discontinuous Galerkin fnite element methods for compressible fuid fows. Int. J. Numer. Anal. Model. 9(4), 928-949 (2012)
22. Giles, M., Pierce, N.:Adjoint equations in CFD-duality, boundary conditions and solution behaviour. In:13th Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics (1997)
23. Giles, M., Süli, E.:Adjoint methods for PDEs:a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145-236 (2002)
24. Harriman, K., Gavaghan, D.J., Süli, E.:The importance of adjoint consistency in the approximation of linear functionals using the discontinuous Galerkin fnite element method. Oxford University Computing Laboratory, Tech. rep. (2004)
25. Hartmann, R.:The role of the Jacobian in the adaptive discontinuous Galerkin method for the compressible Euler equations. In:Warnecke, G. (ed.) Analysis and Numerics for Conservation Laws, pp. 301-316. Springer, Berlin (2005)
26. Hartmann, R.:Derivation of an adjoint consistent discontinuous Galerkin discretization of the compressible Euler equations. In:Lube, G., Papin, G. (eds.) International Conference on Boundary and Interior Layers, Germany (2006)
27. Hartmann, R.:Adjoint consistency analysis of discontinuous Galerkin discretizations. SIAM J. Numer. Anal. 45(6), 2671-2696 (2007)
28. Hartmann, R., Houston, P.:Adaptive discontinuous Galerkin fnite element methods for the compressible Euler equations. J. Comput. Phys. 183(2), 508-532 (2002)
29. Hartmann, R., Houston, P.:Symmetric interior penalty DG methods for the compressible NavierStokes equations I:method formulation. Int. J. Numer. Anal. Model. 1, 1-20 (2006)
30. Hartmann, R., Houston, P.:Symmetric interior penalty DG methods for the compressible NavierStokes equations Ⅱ:goal-oriented a posteriori error estimation. Int. J. Numer. Anal. Model. 3, 141-162 (2006)
31. Hartmann, R., Leicht, T.:Generalized adjoint consistent treatment of wall boundary conditions for compressible fows. J. Comput. Phys. 300, 754-778 (2015)
32. Leicht, T., Hartmann, R.:Anisotropic mesh refnement for discontinuous Galerkin methods in twodimensional aerodynamic fow simulations. Int. J. Numer. Methods Fluids 56(11), 2111-2138 (2008)
33. Loseille, A., Dervieux, A., Alauzet, F.:Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations. J. Comput. Phys. 229(8), 2866-2897 (2010)
34. Lu, J.:An a posteriori control framework for adaptive precision optimization using discontinuous Galerkin fnite element method. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge (2005)
35. Rangarajan, A., May, G., Dolejsi, V.:Adjoint-based anisotropic hp-adaptation for discontinuous Galerkin methods using a continuous mesh model. J. Comput. Phys. 409, 109321 (2020)
36. Sharbatdar, M., Ollivier-Gooch, C.:Mesh adaptation using C¹ interpolation of the solution in an unstructured fnite volume solver. Int. J. Numer. Methods Fluids 86(10), 637-654 (2018)
37. Vassberg, J.C., Jameson, A.:In pursuit of grid convergence for two-dimensional Euler solutions. J. Aircr. 47(4), 1152-1166 (2010)
38. Venditti, D., Darmofal, D.:Grid adaptation for functional outputs:application to two-dimensional inviscid fows. J. Comput. Phys. 176(1), 40-69 (2002)
39. Venditti, D., Darmofal, D.:Anisotropic grid adaptation for functional outputs:application to twodimensional viscous fows. J. Comput. Phys. 187(1), 22-46 (2003)
40. Vijayasundaram, G.:Transonic fow simulation using upstream centered scheme of Godunov type in fnite elements. J. Comput. Phys. 63, 416-433 (1986)
41. Yano, M., Darmofal, D.L.:An optimization-based framework for anisotropic simplex mesh adaptation. J. Comput. Phys. 231(22), 7626-7649 (2012)

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