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Keywords:
$hp$-methods; anisotropic mesh adaptation; interpolation error estimates
$hp$-methods; anisotropic mesh adaptation; interpolation error estimates
Summary:
We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken $H^1$-seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of the proposed anisotropic adaptive strategy in comparison with other adaptive approaches.
References:
[1] Ait-Ali-Yahia, D., Baruzzi, G., Habashi, W. G., Fortin, M., Dompierre, J., Vallet, M.-G.: Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver- independent CFD. II. Structured grids. Int. J. Numer. Methods Fluids 39 (2002), 657-673. DOI 10.1002/fld.356 | MR 1911881 | Zbl 1101.76350
[2] Aubry, R., Löhner, R.: Generation of viscous grids at ridges and corners. Int. J. Numer. Methods Eng. 77 (2009), 1247-1289. DOI 10.1002/nme.2446 | Zbl 1156.76432
[3] Babuška, I., Suri, M.: The $p$ and $h$-$p$ versions of the finite element method, basic principles and properties. SIAM Rev. 36 (1994), 578-632. DOI 10.1137/1036141 | MR 1306924 | Zbl 0813.65118
[4] Cao, W.: Anisotropic measures of third order derivatives and the quadratic interpolation error on triangular elements. SIAM J. Sci. Comput. 29 (2007), 756-781. DOI 10.1137/050634700 | MR 2306267 | Zbl 1136.65100
[5] Cao, W.: An interpolation error estimate in $\mathcal R^2$ based on the anisotropic measures of higher order derivatives. Math. Comput. 77 (2008), 265-286. DOI 10.1090/S0025-5718-07-01981-3 | MR 2353953 | Zbl 1149.65010
[6] Clavero, C., Gracia, J. L., Jorge, J. C.: A uniformly convergence alternating direction HODIE finite difference scheme for 2D time-dependent convection-diffusion problems. IMA J. Numer. Anal. 26 (2006), 155-172. DOI 10.1093/imanum/dri029 | MR 2193974
[7] Dawson, C., Sun, S., Wheeler, M. F.: Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193 (2004), 2565-2580. DOI 10.1016/j.cma.2003.12.059 | MR 2055253 | Zbl 1067.76565
[8] Demkowicz, L., Rachowicz, W., Devloo, P.: A fully automatic $hp$-adaptivity. J. Sci. Comput. 17 (2002), 117-142. DOI 10.1023/A:1015192312705 | MR 1910555 | Zbl 0999.65121
[9] Dolejší, V.: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci. 1 (1998), 165-178. DOI 10.1007/s007910050015 | MR 1839196 | Zbl 0917.68214
[10] Dolejší, V.: ANGENER---software package. Charles University Prague, Faculty of Mathematics and Physics, 2000. www.karlin.mff.cuni.cz/ {dolejsi/angen/angen.htm}.
[11] Dolejší, V.: Analysis and application of the IIPG method to quasilinear nonstationary convection-diffusion problems. J. Comput. Appl. Math. 222 (2008), 251-273. DOI 10.1016/j.cam.2007.10.055 | MR 2474628 | Zbl 1165.65055
[12] Dolejší, V.: {$hp$}-DGFEM for nonlinear convection-diffusion problems. Math. Comput. Simul. 87 (2013), 87-118. DOI 10.1016/j.matcom.2013.03.001 | MR 3046879
[13] Dolejší, V.: Anisotropic {$hp$}-adaptive method based on interpolation error estimates in the {$L^q$}-norm. Appl. Numer. Math. 82 (2014), 80-114. DOI 10.1016/j.apnum.2014.03.003 | MR 3212381 | Zbl 1291.65340
[14] Dolejší, V., Felcman, J.: Anisotropic mesh adaptation for numerical solution of boundary value problems. Numer. Methods Partial Differ. Equations 20 (2004), 576-608. DOI 10.1002/num.10104 | MR 2060780 | Zbl 1060.65125
[15] Dolejší, V., Roos, H.-G.: BDF-FEM for parabolic singularly perturbed problems with exponential layers on layers-adapted meshes in space. Neural Parallel Sci. Comput. 18 (2010), 221-235. MR 2722205
[16] Frey, P. J., Alauzet, F.: Anisotropic mesh adaptation for CFD computations. Comput. Methods Appl. Mech. Eng. 194 (2005), 5068-5082. DOI 10.1016/j.cma.2004.11.025 | MR 2163554 | Zbl 1092.76054
[17] John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. I. A review. Comput. Methods Appl. Mech. Eng. 196 (2007), 2197-2215. DOI 10.1016/j.cma.2006.11.013 | MR 2302890 | Zbl 1173.76342
[18] Knopp, T., Lube, G., Rapin, G.: Stabilized finite element methods with shock capturing for advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 191 (2002), 2997-3013. DOI 10.1016/S0045-7825(02)00222-0 | MR 1903196 | Zbl 1001.76058
[19] Laug, P., Borouchaki, H.: BL2D-V2: isotropic or anisotropic 2D mesher. INRIA, 2002. https://www.rocq.inria.fr/gamma/Patrick.Laug/logiciels/bl2d-v2/INDEX.html</b>
[20] Loseille, A., Alauzet, F.: Continuous mesh framework part I: well-posed continuous interpolation error. SIAM J. Numer. Anal. 49 (2011), 38-60. DOI 10.1137/090754078 | MR 2764420
[21] Loseille, A., Alauzet, F.: Continuous mesh framework part II: validations and applications. SIAM J. Numer. Anal. 49 (2011), 61-86. DOI 10.1137/10078654X | MR 2764421
[22] Mirebeau, J.-M.: Optimal meshes for finite elements of arbitrary order. Constr. Approx. 32 (2010), 339-383. DOI 10.1007/s00365-010-9090-y | MR 2677884 | Zbl 1202.65015
[23] Mirebeau, J.-M.: Optimally adapted meshes for finite elements of arbitrary order and {$W^{1,p}$} norms. Numer. Math. 120 (2012), 271-305. DOI 10.1007/s00211-011-0412-1 | MR 2874967 | Zbl 1238.65116
[24] Schwab, C.: $p$- and $hp$-Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation Clarendon Press, Oxford (1998). MR 1695813 | Zbl 0910.73003
[25] Šolín, P.: Partial Differential Equations and the Finite Element Method. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts John Wiley & Sons, Hoboken (2006). MR 2180081
[26] Šolín, P., Demkowicz, L.: Goal-oriented $hp$-adaptivity for elliptic problems. Comput. Methods Appl. Mech. Eng. 193 (2004), 449-468. DOI 10.1016/j.cma.2003.09.015 | MR 2033961 | Zbl 1044.65082
[27] Sun, S.: Discontinuous Galerkin methods for reactive transport in porous media. Ph.D. thesis, The University of Texas, Austin (2003). MR 2705499
[28] Vejchodský, T., Šolín, P., Zítka, M.: Modular {$hp$}-FEM system HERMES and its application to Maxwell's equations. Math. Comput. Simul. 76 (2007), 223-228. DOI 10.1016/j.matcom.2007.02.001 | MR 2392482 | Zbl 1157.78356
[29] Zienkiewicz, O. C., Wu, J.: Automatic directional refinement in adaptive analysis of compressible flows. Int. J. Numer. Methods Eng. 37 (1994), 2189-2210. DOI 10.1002/nme.1620371304 | MR 1285070 | Zbl 0810.76045
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