×

Hamilton-Jacobi equations on an evolving surface. (English) Zbl 1426.65134

Authors’ abstract: We consider the well-posedness and numerical approximation of a Hamilton-Jacobi equation on an evolving hypersurface in \(\mathbb{R}^3\). Definitions of viscosity sub- and supersolutions are extended in a natural way to evolving hypersurfaces and provide uniqueness by comparison. An explicit in time monotone numerical approximation is derived on evolving interpolating triangulated surfaces. The scheme relies on a finite volume discretisation which does not require acute triangles. The scheme is shown to be stable and consistent, leading to an existence proof via the proof of convergence. Finally, an error bound is proved of the same order as in the flat stationary case.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35F21 Hamilton-Jacobi equations
35D40 Viscosity solutions to PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adalsteinsson, David; Sethian, J. A., Transport and diffusion of material quantities on propagating interfaces via level set methods, J. Comput. Phys., 185, 1, 271-288 (2003) · Zbl 1047.76093 · doi:10.1016/S0021-9991(02)00057-8
[2] Alphonse, Amal; Elliott, Charles M.; Stinner, Bj\"{o}rn, An abstract framework for parabolic PDEs on evolving spaces, Port. Math., 72, 1, 1-46 (2015) · Zbl 1323.35103 · doi:10.4171/PM/1955
[3] Alphonse, Amal; Elliott, Charles M.; Stinner, Bj\"{o}rn, On some linear parabolic PDEs on moving hypersurfaces, Interfaces Free Bound., 17, 2, 157-187 (2015) · Zbl 1333.35306 · doi:10.4171/IFB/338
[4] Bardi, Martino; Capuzzo-Dolcetta, Italo, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, xviii+570 pp. (1997), Birkh\"{a}user Boston, Inc., Boston, MA · Zbl 0890.49011 · doi:10.1007/978-0-8176-4755-1
[5] Barles, Guy, An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications. Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math. 2074, 49-109 (2013), Springer, Heidelberg · Zbl 1269.49043 · doi:10.1007/978-3-642-36433-4\_2
[6] Barrett, John W.; Garcke, Harald; N\"{u}rnberg, Robert, On the stable numerical approximation of two-phase flow with insoluble surfactant, ESAIM Math. Model. Numer. Anal., 49, 2, 421-458 (2015) · Zbl 1315.35156
[7] BreDuEll16 T. Bretschneider, C.-J. Du, C. M. Elliott, T. Ranner, and B. Stinner, Solving reaction-diffusion equations on evolving surfaces defined by biological image data, arXiv preprint arXiv:1606.05093, 2016.
[8] Cheng, Li-Tien; Burchard, Paul; Merriman, Barry; Osher, Stanley, Motion of curves constrained on surfaces using a level-set approach, J. Comput. Phys., 175, 2, 604-644 (2002) · Zbl 0996.65013 · doi:10.1006/jcph.2001.6960
[9] Crandall, M. G.; Lions, P.-L., Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp., 43, 167, 1-19 (1984) · Zbl 0556.65076 · doi:10.2307/2007396
[10] Dziuk, G.; Elliott, C. M., Finite elements on evolving surfaces, IMA J. Numer. Anal., 27, 2, 262-292 (2007) · Zbl 1120.65102 · doi:10.1093/imanum/drl023
[11] Dziuk, Gerhard; Elliott, Charles M., Finite element methods for surface PDEs, Acta Numer., 22, 289-396 (2013) · Zbl 1296.65156 · doi:10.1017/S0962492913000056
[12] Dziuk, Gerhard; Kr\`“{o}ner, Dietmar; M\'”{u}ller, Thomas, Scalar conservation laws on moving hypersurfaces, Interfaces Free Bound., 15, 2, 203-236 (2013) · Zbl 1350.35115 · doi:10.4171/IFB/301
[13] Eilks, C.; Elliott, C. M., Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method, J. Comput. Phys., 227, 23, 9727-9741 (2008) · Zbl 1149.76027 · doi:10.1016/j.jcp.2008.07.023
[14] EllStiVen12 C. M. Elliott, B. Stinner, and C. Venkataraman, Modelling cell motility and chemotaxis with evolving surface finite elements, J. Royal Soc. Interface 9 (2012), no. 76, 3027-3044.
[15] Giesselmann, Jan; M\"{u}ller, Thomas, Geometric error of finite volume schemes for conservation laws on evolving surfaces, Numer. Math., 128, 3, 489-516 (2014) · Zbl 1306.65250 · doi:10.1007/s00211-014-0621-5
[16] Giga, Yoshikazu, Surface Evolution Equations, Monographs in Mathematics 99, xii+264 pp. (2006), Birkh\"{a}user Verlag, Basel · Zbl 1096.53039
[17] Jankuhn, Thomas; Olshanskii, Maxim; Reusken, Arnold, Incompressible fluid problems on embedded surfaces: modeling and variational formulations, Interfaces Free Bound., 20, 3, 353-377 (2018) · Zbl 1406.35224 · doi:10.4171/IFB/405
[18] Kim, Kwangil; Li, Yonghai, Convergence of finite volume schemes for Hamilton-Jacobi equations with Dirichlet boundary conditions, J. Comput. Math., 33, 3, 227-247 (2015) · Zbl 1340.65188 · doi:10.4208/jcm.1411-m4406
[19] Kossioris, G.; Makridakis, Ch.; Souganidis, P. E., Finite volume schemes for Hamilton-Jacobi equations, Numer. Math., 83, 3, 427-442 (1999) · Zbl 0938.65089 · doi:10.1007/s002110050457
[20] Kov\'{a}cs, Bal\'{a}zs; Guerra, Christian Andreas Power, Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces, Numer. Methods Partial Differential Equations, 32, 4, 1200-1231 (2016) · Zbl 1351.65065 · doi:10.1002/num.22047
[21] Lengeler, Daniel; M\"{u}ller, Thomas, Scalar conservation laws on constant and time-dependent Riemannian manifolds, J. Differential Equations, 254, 4, 1705-1727 (2013) · Zbl 1263.35160 · doi:10.1016/j.jde.2012.11.002
[22] Lenz, Martin; Nemadjieu, Simplice Firmin; Rumpf, Martin, A convergent finite volume scheme for diffusion on evolving surfaces, SIAM J. Numer. Anal., 49, 1, 15-37 (2011) · Zbl 1254.65095 · doi:10.1137/090776767
[23] Li, Xiang-Gui; Yan, Wei; Chan, C. K., Numerical schemes for Hamilton-Jacobi equations on unstructured meshes, Numer. Math., 94, 2, 315-331 (2003) · Zbl 1029.65111 · doi:10.1007/s00211-002-0418-9
[24] Macdonald, Colin B.; Ruuth, Steven J., Level set equations on surfaces via the closest point method, J. Sci. Comput., 35, 2-3, 219-240 (2008) · Zbl 1203.65143 · doi:10.1007/s10915-008-9196-6
[25] Mantegazza, Carlo; Mennucci, Andrea Carlo, Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Appl. Math. Optim., 47, 1, 1-25 (2003) · Zbl 1048.49021 · doi:10.1007/s00245-002-0736-4
[26] Olshanskii, Maxim A.; Reusken, Arnold, Error analysis of a space-time finite element method for solving PDEs on evolving surfaces, SIAM J. Numer. Anal., 52, 4, 2092-2120 (2014) · Zbl 1307.65120 · doi:10.1137/130936877
[27] Teigen, Knut Erik; Li, Xiangrong; Lowengrub, John; Wang, Fan; Voigt, Axel, A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Commun. Math. Sci., 7, 4, 1009-1037 (2009) · Zbl 1186.35168
[28] Vierling, Morten, Parabolic optimal control problems on evolving surfaces subject to point-wise box constraints on the control-theory and numerical realization, Interfaces Free Bound., 16, 2, 137-173 (2014) · Zbl 1295.49004 · doi:10.4171/IFB/316
[29] Xu, Jian-Jun; Zhao, Hong-Kai, An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput., 19, 1-3, 573-594 (2003) · Zbl 1081.76579 · doi:10.1023/A:1025336916176
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.