1. Applications of the Surface Finite ElementMethodTom Ranner Mathematics Department September 2011ENUMATH 2011, Leicester

2. The ProblemMany processes in biology and uid mechanics are governed bydiusion on a membrane or interface coupled to diusion in anenclosed bulk region. Other approaches to this problem include aboundary integral formulation1 and a nite volume approach2 . Wewish to using the surface nite element method3 along withstandard nite element techniques4 for the bulk region.1Booty and Siegel 20102Novak, Gao, Choi, Resasco, Scha, and Slepchenko 20073Dziuk 1988; Dziuk and Elliott 20074Lenoir 1986 3. Example 1: Turing InstabilitiesGTP-binding proteins (GTPase) molecules are importantregulators in cells that continuously run through anactivation/deactivation andmembrane-attachment/membrane-detachment cycle.Activated GTPase is able to localise in parts of themembranes and to induce cell polarity5 .This can be modelled by Turing instabilities in areaction-diusion system with attachment and detachment. Figure 5 Rho GT Pases and cell migration. Cell migration requires actin-dependent protrusions at the front (red) and contractile actin:myosin laments (re at the rear. In addition, microtubules (gre n) originating fromd)e thecentrosome(purple arepreferentially stabilized in thedirection of migration allowing targeted vesicle) trafcking from the Golgi (b wn) to the leading edge. ro Figure: Example of cell polarisation from6 5 Rtz and Rger 2011ao 6 Jae and Hall 2005 4. Example 2: Surfactant problem Surfactants, or surface contaminants, signicantly alter the interfacial properties of a uid by changing the surface tension. An example is the tip-streaming of thin threads or small droplets from a drop or bubble this is stretched in an imposed extensional ow7 .FIG. 2. Microuidic ow focusing geometry. a Schematic diagram of thedesign denoting ow of both the inner and outer liquids from left to right. b Image of the orice region of an actual microchannel outlined with the dashed line in a , including a typical image of the water-oil interface extending toward the orice from the upstream channel during a ow ex- periment. Dimensions shown are Wup = 280 m, a = 90 m, Z = 180 m,and Wor = 34 m.8 Figure: Taken from 7 Booty and Siegel 2010 8 Anna and Mayer 2006 5. A Model ProblemAs a model problem we will look to solve the following ellipticproblem:We assume we have a bounded domain Rn with Lipschitzboundary . We wish to nd u : R and v : R solution ofthe systemu + u = f in (1a)u (u v ) + = 0 on (1b) v + v (u v ) = g on . (1c)Here we assume , > 0 are given constants and f and g areknown functions on and respectively. We denote by theLaplace-Beltrami operator on . 6. A Model Problem: Weak formWe can convert this to a weak form using integration by parts over and then using the boundary condition and taking anappropriate linear combination of the two resulting forms leads tothe problem:Find u : R and v : R such that u + u + v + v + (u v )( ) = f+ g for all (, ) H 1 () H 1 ().(2) 7. A Model Problem: Existence, uniqueness and regularity Theorem (Existence and uniqueness) If , > 0 and (f , g ) L2 () L2 (), there exists a unique pair (u, v ) H 1 () H 1 () which satises the weak form (2). Proof. Apply the Lax-Milgram theorem in the Hilbert space H 1 () H 1 () := {(, ) : H 1 () and H 1 ()}, with the norm12 2 2(, ) H 1 ()H 1 () := H 1 () + H 1 () 8. A Model Problem: Existence, uniqueness and regularity Theorem (Regularity) Under the extra assumption that C 2 then (u, v ) H 2 () H 2 () and(u, v )H 2 ()H 2 () c (f , g ) L2 ()L2 () . Proof. Apply standard regularity results for bulk domains9 and surface domains10 to the weak form (2) with = 0 and = 0 respectively. 9Gilbarg and Trudinger 1983 10Aubin 1982 9. A Model Problem: Domain Discretisation We dene h to be a polyhedral approximation of , with boundary h = h , such that nodes of h lie on . We take a quasi-uniform triangulation Th of h with simplices and dene h = max{diamT : T Th }.hh With this construction Th |h , the restriction of Th to h is a quasi-uniform triangulation of h . We assume that T h has at most one face of T . 10. A Model Problem: Finite Element Approximation We dene the following nite element spaces: Vh = {h : h R : h |T is linear, for each T Th } Sh = {h : h R : h |e is linear, for each e Th |h }. The discrete problem is: Find (uh , vh ) Vh Sh such that uh h + uh h + h v h h h + v h hhh+ (uh vh )(h h ) = fh h + gh h h h h for all (h , h ) Vh Sh .(3) fh and gh are some approximation of the data and will be specied later. 11. A Model Problem: Numerical example This method was implemented in the ALBERTA nite element toolbox11 , with = {(x, y , z) : x 2 + y 2 /2 + z 2 /3 < 1} with = = 1, f (x, y , z) = 0 and g (x, y , z) = xy . fh and gh are taken as the interpolants of f and g . 11Schmidt, Siebert, Kster, and Heine 2005 o 12. A Model Problem: Numerical example This method was implemented in the ALBERTA nite element toolbox11 , with = {(x, y , z) : x 2 + y 2 /2 + z 2 /3 < 1} with = = 1, f (x, y , z) = 0 and g (x, y , z) = xy . fh and gh are taken as the interpolants of f and g . 11Schmidt, Siebert, Kster, and Heine 2005 o 13. A Model Problem: Abstract form In order to perform error analysis, we introduce the following abstract forms: a() (w , ) = w + wa() (y , ) = y + y a() (w , y ), (, ) =(w y )( ) and l () () = f l () () = g nally, a (w , y ), (, ) = a() (w , ) + a() (y , )+ a() (w , y ), (, ) l (, ) = l () () + l () (). 14. A Model Problem: Abstract form and the following approximate forms:() ah (wh , h ) = wh h + wh h h()ah (yh , h ) = h yh h h + yh h h() ah(wh , yh ), (h , h ) =(wh yh )(h h )h and() ()lh (h ) = fh h lh (h ) = gh h .h h nally, () (h )ah (wh , yh ), (h , h ) = ah (wh , h ) + ah (yh , h )()+ ah(wh , yh ), (h , h ) ()() lh (h , h ) = lh (h ) + lh (h ). 15. A Model Problem: Surface Lift Since the exact problem and the discrete problem are posed on dierent domains, we must relate the two domains. We start with the surface.We use normal projection of the p(x) d(x) (p(x))domain. For h small enough, foreach point x h there exists a xx unique p(x) . Given byh p(x) p(x) = x d(x)(p(x)). 16. A Model Problem: Surface Geometric Estimates These assumptions give us the following result12 . Lemma Let d denote a signed distance function for , thend L (h ) ch2 . If we denote by h the quotient of the measures on the surface and approximate surface, so that do = h doh , we have thatsup |1 h | ch2 . h Let P denote projection onto the tangent space of and Ph projection onto the tangent space of h . We introduce the notation1 Qh = h (I dH)PPh P(I dH) then we have the estimate |I h Qh | ch2 . 12Dziuk 1988; Dziuk and Elliott 2007 17. A Model Problem: Bulk domain perturbation I In order to relate the bulk domains, we introduce an exact triangulation13 of .TFT^T 13Bernardi 1989 14Dubois 1987 18. A Model Problem: Bulk domain perturbation I In order to relate the bulk domains, we introduce an exact triangulation13 of .~hTT~FT^ FT T Dubois gives a construction of such maps FT to triangulate smooth domains 14 , details of which will not be given here. 13Bernardi 1989 14Dubois 1987 19. A Model Problem: Bulk domain perturbation I In order to relate the bulk domains, we introduce an exact triangulation13 of .~T G| hTT~FT ^FT T Dubois gives a construction of such maps FT to triangulate smooth domains 14 , details of which will not be given here. 1 We dene Gh : h locally by Gh |T : FT FT . 13Bernardi 1989 14Dubois 1987 20. A Model Problem: Bulk domain perturbation I In order to relate the bulk domains, we introduce an exact triangulation13 of . ~ B hT Bh G|hT T ~FT^FTT Dubois gives a construction of such maps FT to triangulate smooth domains 14 , details of which will not be given here. 1 We dene Gh : h locally by Gh |T : FT FT . This is a dieomorphism and is the identity when restricted to interior simplices, those with at most one boundary vertex. We call the domain where Gh is dierent from the identity Bh . 13Bernardi 1989 14Dubois 1987 21. A Model Problem: Bulk domain perturbation II Lemma (15 ) Let T Th be a boundary simplex and T the associated exact triangle. We denote by Jh |T the absolute value of the determinant of DGh |T . Under the assumption that Th is quasi-uniform and C 2 , then for suciently small h, we have that DGh |T I L (T ) chD 2 Gh |T I L (T )c J h |T I L (T ) ch, for some constant independent of T and h. Furthermore,|Bh | ch2 , for some constant independent of h. 15Lenoir 1986 22. A Model Problem: Lifted functionsFor h Vh we dene its lift h : R by h (Gh (x)) := h (x).For h Sh we dene its lift h : R by h (p(x)) := h (x).We also dene the lifted nite element functionsVh = {h : h Vh }Sh = {h : h Sh }.In the analysis, we will assume fh = f and gh = g toavoid smoothness requirements. 23. A Model Problem: Error Bounds Theorem Let (u, v ) H 2 () H 2 () be the solution of the variational problem (2) and let (uh , vh ) Vh Sh be the solution of the nite element scheme given by (3). Denote by uh and vh the lifts of uh and vh respectively. Then we have the following error bound:(u uh , v vh ) C1 h H 1 ()H 1 () where C1 = c(u, v ) H 2 ()H 2 () + (f , g ) L2 ()L2 () . 24. A Model Problem: Error Bounds (cont.) Theorem (cont.) Furthermore, if f L () and u W 1, () then(u uh , v vh ) C2 h 2L2 ()L2 () where C2 = c f L () + g L2 () + uW 1, () + u H 2 () + v H 2 () . 25. A Model Problem: Proof of error bounds I Lemma (Approximation property16 ) For the lifted nite element spaces Vh and Sh , there exists an interpolation operator Ih : H 2 () H 2 () Vh Sh such thatw Ih wL2 ()L2 () +h w Ih w H 1 ()H 1 () ch2 w H 2 ()H 2 () for all w H 2 () H 2 (). 16Dziuk 1988; Lenoir 1986 26. A Model Problem: Proof of error bounds I Lemma (Approximation property16 ) For the lifted nite element spaces Vh and Sh , there exists an interpolation operator Ih : H 2 () H 2 () Vh Sh such thatw Ih wL2 ()L2 () +hw Ih w H 1 ()H 1 () ch2 w H 2 ()H 2 () for all w H 2 () H 2 (). Lemma (Bulk domain errors) Let wh , h Vh and denote their lifts by wh , h then() a() (wh , h ) ah (wh , h ) ch whhH 1 ()H 1 ()() l () (h ) lh (h ) ch fL2 () h . L2 () 16Dziuk 1988; Lenoir 1986 27. A Model Problem: Proof of error bounds II Lemma (Surface domain errors) Let (wh , yh ), (h , h ) Vh Sh and let yh , h denote the lifts of yh , h respectively and wh , h denote the lifts of the traces of wh , h . Then()a() (yh , h ) ah (yh , h ) ch2 yhh H 1 ()H 1 () ()a() (wh , yh ), (h , h ) ah(wh , yh ), (h , h ) ch2 (wh , yh ) (h , h )L2 ()L2 ()L2 ()L2 () ()l () (h ) lh (h ) ch2 f L2 () h. L2 () 28. A Model Problem: Domain approximation errors III Lemma Under the extra assumptions that f L () and w W 1, () with w the inverse lift of w onto h . Let h Vh and denote its lifts by h then ()a() (w , h ) ah (w , h ) ch2 w W 1, () h H 1 () ()l () (h ) lh (h ) ch2 fL ()h. L2 () 29. A Model Problem: Proof H 1 error bound I Notice for (h , h ) Vh Sh , with lifts (h , h )Fh (h , h ) := a (u uh , v vh ), (h , h )= l (h , h ) a (uh , vh ), (h , h )= l (h , h ) lh (h , h )+ ah (uh , vh ), (h , h ) a (uh , vh ), (h , h ). Hence Fh h , h ) C1 h h , h . H 1 ()H 1 () 30. A Model Problem: Proof of H 1 error bound II To prove the H 1 error bound, we rewrite the error as a (u uh , v vh ), (u uh , v vh ) = a (u uh , v vh ), (u, v ) Ih (u, v )+ a (uh , v vh ), Ih (u, v ) (uh , vh ) = a (u uh , v vh ), (u, v ) Ih (u, v )+ Fh Ih (u, v ) (uh , vh ) . The result follows from the approximation property and the domain error results. 31. A Model Problem: Proof L2 error bound I To show the L2 bound we start by setting up a dual problem for L2 () L2 (): Find w H 1 () H 1 () such thata (, ), w ) = , (, ) L2 ()L2 () . We assume this has a unique solution with the following regularity resultw H 2 ()H 2 () c L2 ()L2 () . 32. A Model Problem: Proof L2 error bound II If we further assume that f L () and u W 1, (), then we achieve the improved boundFh (h , h ) C2 h2 (h , h ) . H 1 ()H 1 () This follows from applying the improved domain error results with the bound |Bh | ch2 . 33. A Model Problem: Proof L2 error bound III We start by writing the errore = (u uh , v vh ) as the data for the dual problem and test with e so that2e L2 ()L2 () = a(e, we ). Applying the interpolation theory, the H 1 bound, and the improved bound on Fh leads to the L2 bound. 34. A Model Problem: Numerical Results I To test the convergence rate, we applied this method with = = 1 on the unit ball using the ALBERTA nite element toolbox17 and the PARDISO numerical solver18 . The data was chosen so that the exact solution wasu(x, y , z) = xyz andv (x, y , z) = (3 + )xyz. We calculate the right hand side by setting fh and gh to be the interpolants of f and g in Vh and Sh . 17Schmidt, Siebert, Kster, and Heine 2005 o 18Schenk, Waechter, and Hagemann 2007; Schenk, Bollhofer, and Roemer 2006 35. A Model Problem: Errors hL2 erroreocH 1 erroreoc8.201523e-01 2.817753e-02-2.586266e-01 -4.799888e-01 8.965317e-032.137583 1.517243e-01 0.9955072.555341e-01 2.362206e-032.115725 7.961819e-02 1.0228671.321787e-01 5.989949e-042.081457 4.024398e-02 1.0350146.736035e-02 1.502987e-042.051078 2.016042e-02 1.025427 Table: Bulk errors: u uh hL2 erroreocH 1 erroreoc8.201523e-01 2.218139e-01-1.762399e+004.799888e-01 6.606584e-022.260828 9.810869e-01 1.0934112.555341e-01 1.720835e-022.133950 5.028356e-01 1.0602641.321787e-01 4.346577e-032.087386 2.529537e-01 1.0422576.736035e-02 1.089298e-032.052899 1.266557e-01 1.026162Table: Surface errors:v vh 36. More realistic coupling Langmuir Kinetics I In many of the problems we consider, Langmuir kinetics govern the coupling between bulk and surface concentrations. We consider a bulk concentration u and a surface concentration v . We assume the following simple boundary condition19 The net ux across the interface is the net absorbtion minus desorption rates. This is often interpreted by the adsorption-desroption ux on the surface given byo v on u(v v ), where o is a rate of desorption, on is a rate of adsorption, and v is a maximum desired surface concentration20 . 19Kwon and Derby 2001 20Georgievskii, Medvedev, and Stuchebrukhov 2002 37. More realistic coupling Langmuir Kinetics II Sometimes we wish to impose v as a maximum more strongly by replacing the ux by o v on u(v v )+ , where ()+ denotes the positive part. We can also consider the case where the surface concentration is far from saturation o v on u, which is the linear case we have previously analysed. 38. More Realistic Coupling Numerical Experiments To test the dierent coupling models, we have implement the following parabolic system. We wish to nd u : [0, T ] R and v : [0, T ] R such that with q given by,t u u = 0 q = 0, u q = u v ,q +=0 q = u(1 v ) v ,t v v + q = 0.q = u(1 v )+ v We implemented this method using the above method to descritse in space and semi-implicit time steping. 39. More Realistic Coupling Numerical Experiments Figure: From left to right: q = 0, q = u v , q = u(1 v ) v , q = u(1 v )+ v 40. Turing Instabilities: Modelling We look to model the concentrations of active and inactive concentrations of GTPase on a cell membrane and in the cystosol contained within21 . We denote by the cytosolic volume of the cell and = the cell membrane. We look for a bulk concentration u : R of inactive GTPase, and surface concentrations, v1 , v2 : R, of active and inactive GTPase, respectively. We model the system using a reaction-diusion-attachment/detachment system. For the reaction kinetics we assume simplied MichaelisMenten type law for catalysed reactions and a Langmuir rate law for the attachment/detachment. 21Rtz and Rger 2011 ao 41. Turing Instabilities: Equations Nondimensionalisation leads to the following system. t u = Duin t v1 = d1 v1 + f (v1 , v2 )on t v2 = d2 v1 f (v1 , v2 ) + q(u, v1 , v2 ) on wherev1v1 f (v1 , v2 ) =a1 + (a3 a1 ) v2 a4 , a2 + v1 a5 + v1 with the ux conditionD u = q(u, v1 , v2 ) on withq(u, v1 , v2 ) = u(1 (v1 + v2 ))+ v2 . 42. Turing Instabilities: Weak Form Using the same techniques as for the elliptic equation this leads to the weak form:Find (u, v1 , v2 ) H 1 () H 1 () H 1 () such that t u + D u + t v2 + d2 v2 + (u(1 (v1 + v2 ))+ v2 )( ) = f (v1 , v2 ) t v1 + d1 v1 v2 = f (v1 , v2 ), for all (, ) H 1 () H 1 (). 43. Turing Instabilities: Discretisation We use a the above coupled bulk-surface nite element method to discretize in space. We discretise in time by treating the linear diusion terms implicitly and the nonlinear reaction and attachment/detachment terms explicitly. Parameter choices: Diusion Coecients: D = 1000.0, d1 = 1.0, d2 = 1000.0 Reaction Coecients: a1 = 0.0, a2 = 20.0, a3 = 160.0, a4 = 1.0, a5 = 0.5 Nondimensionalised constant: = 400.0 Attachment/Detachment Coecients: = 0.1, = 1.0 Discretisation: = 1.0e 3, h = 0.137025. 44. Turing Instabilities: Numerical ResultsFigure: Left: u, middle: v1 , right: v2 45. Turing Instabilities: Numerical ResultsFigure: Left: u, middle: v1 , right: v2 46. Turing Instabilities: Numerical Results Final Frame Figure: Left: u (colour rescaled), middle: v1 , right: v2 , at nal time 47. ConclusionWe have developed a computational method for solvingcoupled bulk-surface partial dierential equation.We have analysed a model problem and derived optimal ordererror estimates.We have applied the method to areaction-diusion-attachment/detachment problem from cellbiology.In the future, we hope to perform analysis on thereaction-diusion-attachment/detachment problem and lookto derive error estimates.We also hope to look at time dependent domains. 48. References I Shelley L. Anna and Hans C. Mayer. Microscale tipstreaming in a microuidic ow focusing device. Physics of Fluids, 18(12): 121512, 2006. Thierry Aubin. Nonlinear analysis on manifolds, Monge-Ampere equations. Springer, 1982. Christine Bernardi. Optimal Finite-Element Interpolation on Curved Domains. SIAM J. Numer. Anal., 26(5):12121240, October 1989. Michael R. Booty and Mchael Siegel. A hybrid numerical method for interfacial uid ow with soluble surfactant. J. Comput. Phys., 229(10):38643883, May 2010. Francois Dubois. Discrete vector potential representation of a divergence-free vector eld in three-dimensional domains : numerical analysis of a model problem. SIAM J. Numer. Anal., 27(5):11031141, 1987. 49. References II Gerhard Dziuk. Finite elements for the Beltrami operator on arbitrary surfaces. In S Hildebrandt and R Leis, editors, Partial Dierential Equations and Calculus of Variations, volume 1357 of Lecture Notes in Mathematics 1357, pages 142155. Springer, 1988. Gerhard Dziuk and Charles M. Elliott. Surface nite elements for parabolic equations. J. Comput. Math., 25(4):385407, 2007. Yuri Georgievskii, Emile S. Medvedev, and Alexei a. Stuchebrukhov. Proton transport via coupled surface and bulk diusion. The Journal of Chemical Physics, 116(4):1692, 2002. David Gilbarg and Neil S. Trudinger. Elliptic partial dierential equations of second order. Springer-Verlag, 1983. Aron B Jae and Alan Hall. Rho GTPases: biochemistry and biology. Annual review of cell and developmental biology, 21: 24769, January 2005. 50. References III Yong-Il Kwon and Jerey J. Derby. Modeling the coupled eects of interfacial and bulk phenomena during solution crystal growth. Journal of Crystal Growth, 230:328335, 2001. Michel Lenoir. Optimal Isoparametric Finite Elements and Error Estimates for Domains Involving Curved Boundaries. SIAM J. Numer. Anal., 23(3):562580, October 1986. Igor L. Novak, Fei Gao, Yung-Sze Choi, Diana Resasco, James C Scha, and Boris M Slepchenko. Diusion on a Curved Surface Coupled to Diusion in the Volume: Application to Cell Biology. J. Comput. Phys., 226(2):12711290, October 2007. Andreas Rtz and Matthias Rger. Turing instabilities in a a o mathematical model for signaling networks. Arxiv preprint arXiv:1107.1594, pages 121, 2011. Olaf Schenk, Matthias Bollhofer, and Rudolf A. Roemer. On large-scale diagonalization techniques for the Anderson model of localization. SIAM Rev., 28(3):963983, 2006. 51. References IV Olaf Schenk, Andreas Waechter, and Michael Hagemann. Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization. Computat. Optim. Appl., 36(23):321341, 2007. Alfred Schmidt, Kunibert G. Siebert, Daniel Kster, ando Claus-Justus Heine. Design of adaptive nite element software: the nite element toolbox ALBERTA. Springer-Verlag, Berlin-Heidelberg, 2005.