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Tensor FEM for spectral fractional diffusion. (English) Zbl 1429.65272

Summary: We design and analyze several finite element methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains \(\varOmega \subset \mathbb{R}^d\) with \(d=1,2\). For the solution to the Caffarelli-Silvestre extension, we establish analytic regularity with respect to the extended variable \(y\in (0,\infty )\). Specifically, the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to \(y\), taking values in corner-weighted Kondrat’ev-type Sobolev spaces in \(\varOmega\). In \(\varOmega \subset \mathbb{R}^2\), we discretize with continuous, piecewise linear, Lagrangian FEM (\(P_1\)-FEM) with mesh refinement near corners and prove that the first-order convergence rate is attained for compatible data \(f\in \mathbb{H}^{1-s}(\varOmega )\) with \(0<s<1\) denoting the fractional power. We also prove that tensorization of a \(P_1\)-FEM in \(\varOmega\) with a suitable \(hp\)-FEM in the extended variable achieves log-linear complexity with respect to \({\mathscr{N}}_\varOmega\), the number of degrees of freedom in the domain \(\varOmega\). In addition, we propose a novel, sparse tensor product FEM based on a multilevel \(P_1\)-FEM in \(\varOmega\) and on a \(P_1\)-FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to \({\mathscr{N}}_\varOmega\). Finally, under the stronger assumption that the data be analytic in \(\overline{\varOmega}\), and without compatibility at \(\partial \varOmega\), we establish exponential rates of convergence of \(hp\)-FEM for spectral fractional diffusion operators in energy norm. This is achieved by a combined tensor product \(hp\)-FEM for the Caffarelli-Silvestre extension in the truncated cylinder \(\varOmega\times (0,\mathscr{Y})\) with anisotropic geometric meshes that are refined toward \(\partial \varOmega\). We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods to other problem classes and to other boundary conditions on \(\partial \varOmega \).

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35R11 Fractional partial differential equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

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[1] Abramowitz, M., Stegun, I.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964) · Zbl 0171.38503
[2] Acosta, G., Borthagaray, J.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55(2), 472-495 (2017). https://doi.org/10.1137/15M1033952. · Zbl 1359.65246 · doi:10.1137/15M1033952
[3] Adams, R.: Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). Pure and Applied Mathematics, Vol. 65 · Zbl 0314.46030
[4] Ahlfors, L.: Complex analysis, third edn. McGraw-Hill Book Co., New York (1978). An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics · Zbl 1477.30001
[5] Apel, T.: Interpolation of non-smooth functions on anisotropic finite element meshes. M2AN Math. Model. Numer. Anal. 33(6), 1149-1185 (1999). https://doi.org/10.1051/m2an:1999139. · Zbl 0984.65113 · doi:10.1051/m2an:1999139
[6] Apel, T., Melenk, J.: Interpolation and quasi-interpolation in \[h\] h- and \[hp\] hp-version finite element spaces. In: E. Stein, R. de Borst, T. Hughes (eds.) Encyclopedia of Computational Mechanics, second edn., pp. 1-33. John Wiley & Sons, Chichester, UK (2018). Extended preprint at http://www.asc.tuwien.ac.at/preprint/2015/asc39x2015.pdf.
[7] Aurada, M., Feischl, M., Führer, T., Karkulik, M., Praetorius, D.: Energy norm based error estimators for adaptive BEM for hypersingular integral equations. Appl. Numer. Math. 95, 15-35 (2015). https://doi.org/10.1016/j.apnum.2013.12.004. · Zbl 1320.65184 · doi:10.1016/j.apnum.2013.12.004
[8] Babuška, I., Guo, B.: The \[h\] h-\[p\] p version of the finite element method for domains with curved boundaries. SIAM J. Numer. Anal. 25(4), 837-861 (1988). https://doi.org/10.1137/0725048. · Zbl 0655.65124 · doi:10.1137/0725048
[9] Băcuţă, C., Li, H., Nistor, V.: Differential operators on domains with conical points: precise uniform regularity estimates. Rev. Roumaine Math. Pures Appl. 62(3), 383-411 (2017) · Zbl 1399.35171
[10] Bernardi, C., Dauge, M., Maday, Y.: Polynomials in the Sobolev world (version 2). Tech. Rep. 14, IRMAR (2007). https://hal.archives-ouvertes.fr/hal-00153795.
[11] Birman, M., Solomjak, M.: Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve. Leningrad. Univ., Leningrad (1980)
[12] Bonito, A., Borthagaray, J.P., Nochetto, R.H., Otárola, E., Salgado, A.J.: Numerical methods for fractional diffusion. Computing and Visualization in Science (2018). https://doi.org/10.1007/s00791-018-0289-y. · Zbl 07704543
[13] Bonito, A., Pasciak, J.: Numerical approximation of fractional powers of elliptic operators. Math. Comp. 84(295), 2083-2110 (2015). https://doi.org/10.1090/S0025-5718-2015-02937-8. · Zbl 1331.65159 · doi:10.1090/S0025-5718-2015-02937-8
[14] Brändle, C., Colorado, E., de Pablo, A., Sánchez, U.: A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 143(1), 39-71 (2013). https://doi.org/10.1017/S0308210511000175. · Zbl 1290.35304 · doi:10.1017/S0308210511000175
[15] Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions. Trans. Amer. Math. Soc. 367(2), 911-941 (2015). https://doi.org/10.1090/S0002-9947-2014-05906-0. · Zbl 1317.35280 · doi:10.1090/S0002-9947-2014-05906-0
[16] Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052-2093 (2010). https://doi.org/10.1016/j.aim.2010.01.025. · Zbl 1198.35286 · doi:10.1016/j.aim.2010.01.025
[17] Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Part. Diff. Eqs. 32(7-9), 1245-1260 (2007). https://doi.org/10.1080/03605300600987306. · Zbl 1143.26002 · doi:10.1080/03605300600987306
[18] Caffarelli, L., Stinga, P.: Fractional elliptic equations, Caccioppoli estimates and regularity. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(3), 767-807 (2016). https://doi.org/10.1016/j.anihpc.2015.01.004. · Zbl 1381.35211 · doi:10.1016/j.anihpc.2015.01.004
[19] Capella, A., Dávila, J., Dupaigne, L., Sire, Y.: Regularity of radial extremal solutions for some non-local semilinear equations. Comm. Partial Differential Equations 36(8), 1353-1384 (2011). https://doi.org/10.1080/03605302.2011.562954. · Zbl 1231.35076 · doi:10.1080/03605302.2011.562954
[20] Cartan, H.: Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes. Avec le concours de Reiji Takahashi. Enseignement des Sciences. Hermann, Paris (1961) · Zbl 0094.04401
[21] Chen, X., Zeng, F., Karniadakis, G.: A tunable finite difference method for fractional differential equations with non-smooth solutions. Comput. Methods Appl. Mech. Engrg. 318, 193-214 (2017). https://doi.org/10.1016/j.cma.2017.01.020. · Zbl 1439.65082 · doi:10.1016/j.cma.2017.01.020
[22] Costabel, M., Dauge, M.: General edge asymptotics of solutions of second-order elliptic boundary value problems. I, II. Proc. Roy. Soc. Edinburgh Sect. A 123(1), 109-155, 157-184 (1993). https://doi.org/10.1017/S0308210500021272. · Zbl 0791.35032
[23] Costabel, M., Dauge, M., Nicaise, S.: Analytic regularity for linear elliptic systems in polygons and polyhedra. Math. Meths. Appl. Sci. 22(8) (2012) · Zbl 1257.35056
[24] D’Elia, M., Gunzburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66(7), 1245 - 1260 (2013). https://doi.org/10.1016/j.camwa.2013.07.022. http://www.sciencedirect.com/science/article/pii/S0898122113004707. · Zbl 1345.35128
[25] DeVore, R., Lorentz, G.: Constructive Approximation. Springer Verlag. Berlin (1993) · Zbl 0797.41016
[26] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.19 of 2018-06-22. http://dlmf.nist.gov/. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds.
[27] Duoandikoetxea, J.: Fourier analysis, Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence, RI (2001). Translated and revised from the 1995 Spanish original by David Cruz-Uribe · Zbl 0969.42001
[28] Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Comm. Part. Diff. Eqs. 7(1), 77-116 (1982). https://doi.org/10.1080/03605308208820218. · Zbl 0498.35042 · doi:10.1080/03605308208820218
[29] Gaspoz, F., Heine, C.J., Siebert, K.: Optimal grading of the newest vertex bisection and \[H^1\] H1-stability of the \[L_2\] L2-projection. IMA J. Numer. Anal. 36(3), 1217-1241 (2016). https://doi.org/10.1093/imanum/drv044. · Zbl 1433.65291 · doi:10.1093/imanum/drv044
[30] Gaspoz, F., Morin, P.: Convergence rates for adaptive finite elements. IMA J. Numer. Anal. 29(4), 917-936 (2009). https://doi.org/10.1093/imanum/drn039. · Zbl 1183.65134 · doi:10.1093/imanum/drn039
[31] Gold́shtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Amer. Math. Soc. 361(7), 3829-3850 (2009). https://doi.org/10.1090/S0002-9947-09-04615-7. · Zbl 1180.46022
[32] Gui, W., Babuška, I.: The \[h,\;p\] h,p and \[h\] h-\[p\] p versions of the finite element method in \[11\] dimension. II. The error analysis of the \[h\] h- and \[h\] h-\[p\] p versions. Numer. Math. 49(6), 613-657 (1986). https://doi.org/10.1007/BF01389734. · Zbl 0614.65089 · doi:10.1007/BF01389734
[33] Harbrecht, H., Peters, M., Siebenmorgen, M.: Combination technique based \[k\] k-th moment analysis of elliptic problems with random diffusion. J. Comput. Phys. 252, 128-141 (2013). https://doi.org/10.1016/j.jcp.2013.06.013. · Zbl 1349.35441 · doi:10.1016/j.jcp.2013.06.013
[34] Khoromskij, B., Melenk, J.: Boundary concentrated finite element methods. SIAM J. Numer. Anal. 41(1), 1-36 (2003) · Zbl 1050.65113
[35] Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25(3), 537-554 (1984) · Zbl 0557.46025
[36] Landkof, N.: Foundations of modern potential theory. Springer-Verlag, New York-Heidelberg (1972). Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180 · Zbl 0253.31001
[37] Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York (1972) · Zbl 0223.35039
[38] Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial difference equations by tensor product methods. Numer. Math. 6, 185-199 (1964). https://doi.org/10.1007/BF01386067. · Zbl 0126.12703 · doi:10.1007/BF01386067
[39] McLean, W.: Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000) · Zbl 0948.35001
[40] Meidner, D., Pfefferer, J., Schürholz, K., Vexler, \[B.: hp\] hp-finite elements for fractional diffusion. SIAM Journal on Numerical Analysis 56(4), 2345-2374 (2018). https://doi.org/10.1137/17M1135517. · Zbl 1397.65282 · doi:10.1137/17M1135517
[41] Melenk, J.: On the robust exponential convergence of \[hp\] hp finite element method for problems with boundary layers. IMA J. Numer. Anal. 17(4), 577-601 (1997). https://doi.org/10.1093/imanum/17.4.577. · Zbl 0887.65106 · doi:10.1093/imanum/17.4.577
[42] Melenk, J.: \[hp\] hp-finite element methods for singular perturbations, Lecture Notes in Mathematics, vol. 1796. Springer-Verlag, Berlin (2002). https://doi.org/10.1007/b84212. · Zbl 1021.65055
[43] Melenk, J., Schwab, \[C.: hp\] hp FEM for reaction-diffusion equations. I. Robust exponential convergence. SIAM J. Numer. Anal. 35(4), 1520-1557 (1998). https://doi.org/10.1137/S0036142997317602. · Zbl 0972.65093 · doi:10.1137/S0036142997317602
[44] Melenk, J., Schwab, C.: Analytic regularity for a singularly perturbed problem. SIAM J. Math. Anal. 30(2), 379-400 (1999). https://doi.org/10.1137/S0036141097317542. · Zbl 1023.35009 · doi:10.1137/S0036141097317542
[45] Miller, K., Samko, S.: Completely monotonic functions. Integral Transform. Spec. Funct. 12(4), 389-402 (2001). https://doi.org/10.1080/10652460108819360. · Zbl 1035.26012 · doi:10.1080/10652460108819360
[46] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165, 207-226 (1972) · Zbl 0236.26016
[47] Müller, F., Schötzau, D., Schwab, C.: Symmetric interior penalty discontinuous Galerkin methods for elliptic problems in polygons. SIAM J. Numer. Anal. 55(5), 2490-2521 (2017). https://doi.org/10.1137/17M1120634. · Zbl 1376.65145 · doi:10.1137/17M1120634
[48] Nochetto, R., Otárola, E., Salgado, A.: A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15(3), 733-791 (2015). https://doi.org/10.1007/s10208-014-9208-x. · Zbl 1347.65178 · doi:10.1007/s10208-014-9208-x
[49] Nochetto, R., Otárola, E., Salgado, A.: Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications. Numer. Math. 132(1), 85-130 (2016). https://doi.org/10.1007/s00211-015-0709-6. · Zbl 1334.65030 · doi:10.1007/s00211-015-0709-6
[50] Nochetto, R., Veeser, A.: Primer of adaptive finite element methods. In: Multiscale and adaptivity: modeling, numerics and applications, Lecture Notes in Math., vol. 2040, pp. 125-225. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-24079-9. · Zbl 1252.65192
[51] Olshanskii, M., Reusken, A.: On the convergence of a multigrid method for linear reaction-diffusion problems. Computing 65(3), 193-202 (2000). https://doi.org/10.1007/s006070070006. · Zbl 0972.65082 · doi:10.1007/s006070070006
[52] Otárola, E.: A PDE approach to numerical fractional diffusion. Ph.D. thesis, University of Maryland, College Park (2014)
[53] Roos, H.G., Stynes, M., Tobiska, L.: Numerical methods for singularly perturbed differential equations, Springer Series in Computational Mathematics, vol. 24. Springer-Verlag, Berlin (1996). https://doi.org/10.1007/978-3-662-03206-0. Convection-diffusion and flow problems · Zbl 0844.65075
[54] Sauter, S., Schwab, C.: Boundary element methods, Springer Series in Computational Mathematics, vol. 39. Springer-Verlag, Berlin (2011). https://doi.org/10.1007/978-3-540-68093-2. Translated and expanded from the 2004 German original · Zbl 1215.65183
[55] Schneider, R.: Multiskalen- und Wavelet-Matrixkompression. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (1998). https://doi.org/10.1007/978-3-663-10851-1. Analysisbasierte Methoden zur effizienten Lösung großer vollbesetzter Gleichungssysteme. [Analysis-based methods for the efficient solution of large nonsparse systems of equations] · Zbl 0899.65063
[56] Schneider, R., Reichmann, O., Schwab, C.: Wavelet solution of variable order pseudodifferential equations. Calcolo 47(2), 65-101 (2010). https://doi.org/10.1007/s10092-009-0012-y. · Zbl 1202.65013 · doi:10.1007/s10092-009-0012-y
[57] Schötzau, D., Schwab, C.: Exponential convergence for \[hp\] hp-version and spectral finite element methods for elliptic problems in polyhedra. M3AS 25(9), 1617-1661 (2015) · Zbl 1322.65099
[58] Schötzau, D., Schwab, C.: Exponential convergence of hp-fem for elliptic problems in polyhedra: Mixed boundary conditions and anisotropic polynomial degrees. Journ. Found. Comput. Math. 18(3), 595-660 (2018). https://doi.org/10.1007/s10208-017-9349-9. · Zbl 1402.65166 · doi:10.1007/s10208-017-9349-9
[59] Schwab, \[C.: p\] p- and \[hp\] hp-finite element methods. Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York (1998). Theory and applications in solid and fluid mechanics · Zbl 0910.73003
[60] Schwab, C., Suri, M.: The \[p\] p and \[hp\] hp versions of the finite element method for problems with boundary layers. Math. Comp. 65(216), 1403-1429 (1996). https://doi.org/10.1090/S0025-5718-96-00781-8. · Zbl 0853.65115 · doi:10.1090/S0025-5718-96-00781-8
[61] Schwab, C., Suri, M., Xenophontos, C.: The \[hp\] hp finite element method for problems in mechanics with boundary layers. Comput. Methods Appl. Mech. Engrg. 157(3-4), 311-333 (1998). https://doi.org/10.1016/S0045-7825(97)00243-0. Seventh Conference on Numerical Methods and Computational Mechanics in Science and Engineering (NMCM 96) (Miskolc) · Zbl 0959.74073
[62] Scott, L., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483-493 (1990). https://doi.org/10.2307/2008497. · Zbl 0696.65007 · doi:10.2307/2008497
[63] Stinga, P., Torrea, J.: Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Differential Equations 35(11), 2092-2122 (2010). https://doi.org/10.1080/03605301003735680. · Zbl 1209.26013 · doi:10.1080/03605301003735680
[64] Tartar, L.: An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana, vol. 3. Springer, Berlin (2007) · Zbl 1126.46001
[65] Turesson, B.: Nonlinear potential theory and weighted Sobolev spaces, Lecture Notes in Mathematics, vol. 1736. Springer-Verlag, Berlin (2000). https://doi.org/10.1007/BFb0103908. · Zbl 0949.31006
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