On convergence of the immersed boundary method for elliptic interface problems

Transcription

1 On convergence of te immersed boundary metod for elliptic interface problems Zilin Li January 26, 2012 Abstract Peskin s Immersed Boundary (IB) metod is one of te most popular numerical metods for many years and as been applied to problems in matematical biology, fluid mecanics, material sciences, and many oter areas. Peskin s IB metod is associated wit discrete delta functions. It is believed tat te IB metod is first order accurate in te L norm. But almost no rigorous proof could be found in te literature until recently [14] in wic te autor sowed tat te velocity is indeed first order accurate for te Stokes equations wit a periodic boundary condition. In tis paper, we sow first order convergence wit a log factor of te IB metod for elliptic interface problems essentially witout te boundary condition restrictions. Te results sould be applicable to te IB metod for many different situations involving elliptic solvers for Stokes and Navier-Stokes equations. keywords: Immersed Boundary (IB) metod, Dirac delta function, convergence of IB metod, discrete Green function, discrete Green s formula. AMS Subject Classification M12, 65M20. 1 Introduction Since its invention in 1970 s, te Immersed Boundary (IB) metod [15] as been applied almost everywere in matematics, engineering, biology, fluid mecanics, and many many more areas, see for example, [16] for a review and references terein. Te IB metod is not only a matematical modeling tool in wic a complicated boundary condition can be treated as a source distribution but also a numerical metod in wic a discrete delta function is used. Te IB metod is robust, simple, and as been applied to many problems. Center for Researc in Scientific Computation (CRSC) and Department of Matematics, Nort Carolina State University, Raleig, NC 27695, USA 1

2 It is widely believed tat Peskin s IB metod is only first order accurate in te L norm. However, tere was almost no rigorous proof in te literature until recently [14], in wic te autor as proved te first order accuracy of te IB metod for te Stokes equations wit a periodic boundary condition. Te proof is based on some known inequalities between te fundamental solution and te discrete Green function wit a periodic boundary condition for Stokes equations. In [4], te autor sowed tat te pressure obtained from IB metod as O( 1/2 ) order of convergence in te L 2 norm for a 1D model. In [19, 20], te autors designed some level set metods based on discrete delta functions. Wit suitable quadrature formulas in te integral form using te Green functions, te autors sow tat teir approac can get expected accuracy. However, tere are few teoretical proofs on te IB metod for elliptic interface problems wit general boundary conditions. Tis is te main motivation of tis paper. One difficult is tat tere is little known estimates between te fundamental solution and te discrete Green function wit Diriclet or oter boundary conditions on rectangular domains. Compared wit te case of periodic boundary conditions were tere are existing estimates between te discrete Green function and te continuous one [8], tere is almost none for Diriclet and oter boundary conditions. Te main goal of tis paper is to provide a convergence proof for te IB metod for elliptic interface problems wit Diriclet boundary conditions. We will sow tat wit commonly used discrete delta functions tat satisfy te zerot moment condition and first order interpolation property, te IB metod is indeed first order convergent in te L norm wit a log factor. Te key in our proof is to establis a connection between te discrete Green function and te continuous one. Our proof is essentially independent of te boundary conditions and it is valid in 1D, 2D, and 3D cases. Te result sould be applicable for many IB metods involving Stokes and Navier-Stokes solvers. 2 Proof of te convergence of te IB metod in 1D We will give a proof for te 1D model, u = c δ(x α) 0 < x < 1, 0 < α < 1, u(0) = u(1) = 0, (1) first in tis section. Note tat te analytic solution to te equation above is { cx (1 α) if x α, u(x) = cα (1 x) oterwise. (2) Given a uniform Cartesian grid x i = i, i = 0, 1,, n, = 1/n, te IB metod leads to te following system of linear equations, U i 1 2U i + U i+1 2 = cδ (x i α), i = 1, 2,, n 1, (3) were U i is te finite difference approximation of te solution u(x i ), and δ (x i α) is a discrete delta function applied to te grid point x i. In te matrix-vector form, te above finite difference equations can be written as A U = F, were A is te tri-diagonal matrix formed by te discrete Laplacian. 2

3 It is well known tat A is a symmetric positive definite matrix (SPD) and diagonally dominant. Note tat, a discrete delta function as a compact support in te neigborood of interface, tat is, δ (x) 0 only if x W, were W is a constant. Commonly used discrete delta functions include te at discrete delta function (δ at (x) wit W = 1): δ at (x) = { ( x )/ 2, if x <, 0, if x, and Peskin s original discrete cosine delta function (δ cosine (x) wit W = 2) 1 (1 + cos (πx/2)), if x < 2, δ (x) cosine = 4 0, if x 2. (4) (5) see for example, [13]. Note tat, wen we use te at delta function, te result is te same as tat of te IIM for te simple model. Te solution to te finite difference equations is te same as true solution if tere are no round-off errors, tat is u(x i 1 ) 2u(x i ) + u(x i+1 ) 2 = cδ at (x i α), i = 1, 2,, n 1, (6) see for example, [3, 13]. But tis is not te case for oter discrete delta functions. We define te error vector as E = {E i }, were E i = u(x i ) U i. Te local truncation error is defined as T = {T i }, T i = u(x i 1) 2u(x i ) + u(x i+1 ) 2 cδ (x i α). (7) Wit te definition, we ave A U = F, A u = F + T, and terefore A E = T. For te at discrete delta function, we ave T i = 0 for all i s for te simple model. For te cosine discrete delta function or oter discrete delta functions, generally we ave T j = O(1/) for a few grid points neigboring te interface α, see Table 1 on page 19. So te interesting question is: Wy is te IB metod still first order accurate, tat is, E = O()? To answer tis question, we first introduce te following lemma. Lemma 2.1. Let A y = e k and y 0 = y n = 0, were e k is te k-t unit base vector, ten { xi (1 x k ) if i k, y i = x k (1 x i ) oterwise. (8) Te significance of tis lemma is tat te solution is order smaller tan te concentrated source. Proof: We note te following identity A y = e k = δat e k. (9) 3

4 For tis simple case, te IB metod using te at discrete delta function is identical to te IIM, see [13]. Tus from te Immersed Interface Metod, see [3, 13], we know tat y is te exact discrete solution at te grid points of te following boundary value problem wose solution is u = δ(x x k ), 0 < x < 1, u(0) = u(1) = 0, (10) y(x) = { x (1 xk ) if x x k, x k (1 x) oterwise. Tis completes te proof. Note tat y(x). From tis lemma, we ave te following corollary. Corollary 2.2. Let A y = r, ten y(x) W r, were W is te number of non-zero components of r. Te proof is straigtforward from (11) and te fact tat 0 x 1 and 0 1 x 1. Notice tat for a discrete delta function, it sould satisfy at least te zerot moment equation, see [3], tat is δ (x i α) = 1, (12) corresponding to te continuous case δ (x α)dx = 1. teorem. i (11) Now we are ready to prove te main Teorem 2.3. Let u(x) be te solution to (1) and U is te solution obtained from te immersed boundary metod (3) using a discrete delta function δ (x) for (1). Ten U is first order accurate, tat is were C is a constant. E C, (13) Proof: We can decompose te local truncation error into two groups T = T reg + T irreg, (14) were T reg = 0 corresponds to te local truncation errors at regular grid points were δ (x i α) = 0 and te true solution is piecewise linear. Note tat, we ave T i = T reg + T irreg = O + T irreg. (15) On te oter and, we also ave u(x i 1 ) 2u(x i ) + u(x i+1 ) 2 = cδ at (x i α) (16) 4

5 since te finite difference metod using te discrete delta function gives te exact solution at all te grid points. Tus we ave T i = u(x i 1) 2u(x i ) + u(x i+1 ) 2 cδ (x i α) = cδ at (x i α) cδ (x i α) = 0, (17) from te zerot moment equation (12). Tus we ave i T irreg i groups, one wit all positive T i s denoted as T irreg,+ i. Since T irreg,+ = 0. We divide T irreg i into two ; te oter one is all te negatives denoted as T irreg, i i + T irreg, i = 0, T irreg,+ i and T irreg, i must ave te same order of te magnitude O(1/) altoug tose index i are different except tat x i α W is true for all irregular grid points. Because te solution is linear wit c, we ave E = A 1 From te solution expression, we know tat, assuming tat x l α, E l = A 1 = x l i T = ( A 1 T irreg,+ + T irreg, ). (18) T = ( A 1 T irreg,+ + T irreg, ) T irreg,+ i (1 x i ) + j j (1 x j ) T irreg, = x l (1 α) i = O(W ), T irreg,+ i + j T irreg, j + O(W ) after we expand all x i s and x j s at α and since all related x i and x j are witin W distance from te interface α. Te proof for x l > α is similar except tat we need to use te solution for x > α. Tis completes te proof. 3 Proof of te convergence of te IB metod in 2D Te discussion for 2D problems is muc more callenging since te interface is often a curve instead of a point. In [14], te autor as proved te first order accuracy of te IB metod for te Stokes equations wit a periodic boundary condition in 2D based on existing estimates between te discrete Green function and te continuous one in [8]. However, tere are almost no teoretical proof on te IB metod for elliptic interface problems or oter PDES wit general boundary conditions. We will prove tat te result obtained from te IB for te elliptic interface problem wit a Diriclet boundary condition is indeed first order accurate in tis section. 5

6 1 0.9 Ω Ω Ω=Ω + Ω Ω Figure 1: A diagram of a 2D elliptic interface problem. Te interface is. Consider te following 2D elliptic interface problem, u(x, y) = f(x, y) + v(s)δ (x X(s)) (y Y (s)) ds, (x, y) Ω, u(x, y) Ω = u 0 (x, y), (19) were we assume tat f C(Ω), C 1, v(s) C 1. Witout loss of generality, we assume tat Ω is a unit square 0 x, y 1, see Fig. 1 for an illustration. Te problem can be decomposed as te sum of te solutions of te following two problems. Te first one is u 1 (x, y) = f(x, y) u 1 (x, y) Ω = u 0 (x, y), wic is a regular problem wose solution u 1 (x, y)] C 2 (Ω). Te second problem is u 2 (x, y) = v(s)δ (x X(s)) (y Y (s)) ds, u 2 (x, y) Ω = 0. Te solution to te second problem is equivalent to te following problem [ ] u2 u 2 (x, y) = 0, [u 2 ] = 0, = v(s), n u 2 (x, y) Ω = 0. (20) (21) (22) 6

7 Te solution to te original problem is u = u 1 + u 2. Since u 1 is te solution to a regular problem, it is enoug just to consider u 2 (x, y). Tus we will simply use te notation u(x, y) for u 2 (x, y). Peskin s IB metod for te problem includes te following steps: Generate a uniform Cartesian mes x i = i, y j = j, i, j = 0, 1,, n. Here we use a uniform mes for simplicity. We denote Ω as te set of all grid points; and Ω as te grid points on te boundary. Replace te partial derivatives wit te finite difference approximation and use a discrete delta function to spread te singular source to nearby grid points U i 1,j + U i+1,j + U i,j 1 + U i,j+1 4U ij 2 C IB ij = N b k=1 v k δ (x i X k ) δ (y j Y k ) s k, = C IB ij, i, j = 1, 2,, n 1, (23) were (X k, Y k ), k = 1, 2,, N b, is a discretization of te interface, and v k v(s k ), wic we assume it is at least first order approximation, v k = v(s k ) + O(). Solve te finite difference system of equations above to get an approximation solution {U ij }. Tis can be done by calling a fast Poisson solver, say [1]. 3.1 Discrete delta functions and discrete Green functions As a common practice, we assume tat max k {s k } = s O(). In Peksin s IB metod, a discrete delta function is used for two purposes. One is to spread te singular source to te nearby grids. Te oter one is to interpolate a grid function, say te velocity, to get its values on te interface. Tus te discrete delta function used sould satisfies at least te zerot moment condition as described in [3]. Te interpolation using te discrete delta function sould be at least first order accurate, tat is, i,j=1 wic corresponds to v k δ (x i X k ) δ (y j Y k ) s k = 2 N b Ω k=1 v(s)δ (x X(s)) δ (y Y (s)) dsdxdy = v(s)ds + O(), (24) v(s)ds. (25) From Ω u(x, y)δ (x X) δ (y Y ) dxdy = u(x, Y ), we sould also ave te interpolation property, i,j=1 2 u(x i, y j )δ (x i X k ) δ (y j Y k ) = u(x, Y ) + O(). (26) 7

8 In (24) and (26), te error terms depend on te first order derivatives of v(s) and u(x, y), respectively. Te discrete delta function as a compact support, tat is, δ (x i X k ) = 0, if x i X k > W, and δ (y j Y k ) = 0, if y j Y k > W, (27) were x ij = (x i, y j ), and W is a constant. We define te error vector as E = {E ij }, were E ij = u(x i, y j ) U ij. Te local truncation error is defined as T = {T ij }, T ij = u(x i 1, y j ) + u(x i+1 ), y j ) + u(x i, y j 1 ) + u(x i, y j+1 ) 4u(x i, y j ) 2 C IB ij. (28) In te matrix vector form, we ave A U = F, A u = F + T, and terefore A E = T, were A is te matrix formed by te discrete Laplacian. We ave T ij = O( 2 ) at regular grid points were Cij IB = 0. In general, we ave T ij = O(1/) for grid points neigboring te interface except for te correction terms using te Immersed Interface Metod (IIM) [11, 12, 13] for wic we ave Tij IIM = O() at irregular grid points were te interface cuts troug te standard five-point stencil. It is interesting tat te local truncation errors can ave order O(1/) at some grid points, but te global error is still of O(). Tere as to be some kind of cancelations of te errors, wic can be seen from our proof process. Definition 3.1. Let e lm be te unit grid function wose values are zero at all grid points except at x lm = (x l, y m ) were its component is e lm = 1. Te discrete Green function centered at x lm wit omogeneous boundary condition is defined as ( ) G (x ij, x lm ) = (A ) 1 1 e lm 2, G ( Ω, x lm ) = 0, (29) were Ω denotes te grid points on te boundary Ω. Note tat from Remark in [7], we know tat G (x ij, x lm ) is symmetric, ij G (x ij, x lm ) = G ( x lm, x ij ). (30) Te usual discrete Green function, also called a discrete fundamental solution, on te entire integer lattice is defined as 1 g (x ij, x lm ) = 2, if x ij = x lm, (31) 0, oterwise, for all integers i and j, see for example, [2, 6, 7, 10, 14, 17, 18] for more discussions about te discrete Green s functions. Note tat g (x ij, x lm ) is also symmetric, tat is, g (x ij, x lm ) = g (x lm, x ij ). To prepare te convergence proof, we first list some lemmas tat are eiter directly or indirectly used in te convergence proof. Te following lemma is te discrete first Green s formula. Altoug it is not directly used in te proof, it sows ow te discrete summation is related to te integral form; and ow te source distribution is related to te jump conditions. 8

9 Lemma 3.2. Te discrete first Green s formula and an error estimate. Let u(x, y) be te solution to (19). Tus u(x, y) is in te piecewise C 1 (Ω) space, tat is, u(x, y) C 1 (Ω \ ); Assuming tat te distance between and Ω is O(1), tat is, dist(, Ω) O(1), ten we ave were i,j=1 u(x i, y j ) 2 = Ω u ds + O() = v(s)ds + O(), (32) n u(x i, y j ) = u(x i 1, y j ) + u(x i+1 ), y j ) + u(x i, y j 1 ) + u(x i, y j+1 ) 4u(x i, y j ) 2, (33) is te discrete Laplacian using te standard five-point stencil, and te summation is over all te interior grid points. Proof: We first prove te discrete first Green s formula by expanding te summation. After cancelation of interior terms, only boundary terms are left in te summation as follows, u(x i, y j ) 2 = ij = u(x 0, y j ) u(x 1, y j ) j=1 Ω + + j=1 u(x i, y 0 ) u(x i, y 1 ) u ds + O(). n u(x n, y j ) u(x, y j ) + u(x i, y n ) u(x i, y ) On te oter and, by integrating bot sides of te partial differential equation (19) wit f(x, y) = 0 and u 0 (x, y) = 0, we get ( ) udxdy = v(s)δ (x X(s)) (y Y (s)) ds dxdy, Ω Ω or equivalently, Tis completes te proof. Ω u n ds = v(s) ds. Remark 3.3. Te double integral udxdy can be divided into tree parts Ω Ω udxdy = = Ω Ω + ɛ udxdy + u n ds + ɛ Ω ɛ u n ds + udxdy + + ɛ u n ds, Ω ɛ udxdy 9

10 Ω + ɛ Ω ɛ ɛ Ω ɛ + ɛ Figure 2: A diagram of te domain, interface, and integration. see Fig. 2 for an illustration, see also [12]. As ɛ 0, we ave [ ] u lim udxdy = v(s) ds = ds ɛ 0 Ω ɛ n from te partial differential equation (19) wit f(x, y) = 0 and u 0 (x, y) = 0. Tus we get u lim udxdy = ɛ 0 Ω ɛ Ω n ds. Wile Lemma 3.2 is not directly used in our convergence proof, it is easier to illustrate te relation between te boundary integral along Ω and te source distribution along for te first Green formula tan te second Green formula tat is used in te proof. 3.2 Interpolating te discrete delta function We know tat G (x ij, x lm ) defined in (29) is a grid function wit te omogeneous Diriclet boundary condition at te boundary grid points. Te discrete Laplacian G (x ij, x lm ) is zero at all interior grid points except tat it is 1/ 2 at x lm. We can interpolate G (x ij, x lm ) to te entire domain to get an interpolation function G I (x, x lm). We consider any suc an interpolation function tat satisfies te following: G I (x ij, x lm ) = G (x ij, x lm ). G I (x, x lm) C 1 (Ω) H 2 (Ω). G I (x ij, x lm ) = G (x ij, x lm ) = 0, tat is, zero for all i and j except for i = l and j = m. Te interpolation is second or iger order accurate, tat is α G (x ij, x lm ) α G I (x ij, x lm ) C 3 α 1, for α 1 2, (34) 10

11 were te derivatives of G (x ij, x lm ) is defined from finite differences, see [18], and α is te summation notation as used in te literature for te Sobolev spaces. R lm G I (x, x lm)dxdy = O() except for te four neigboring squares centered at x lm on wic R lm G I (x, x lm)dxdy = 1 + O(). We provide a construction of suc an interpolation function in Appendix. Lemma 3.4. Let G I (x, x lm) be an interpolation function of G (x ij, x lm ) tat satisfies te conditions above, ten we ave te following estimates: G I (x, x lm) log ( x x lm ) + O(), (35) α G I (x, x lm ) C ( x ij x lm 2 + ) α 1 + O(), if α 1 k 1, (36) were k is te order of interpolation and C is a constant. Te inequality (36) is true if dist(x, Ω) O(1) and dist( Ω, x lm ) O(1). Proof: From Remark in [7], we know tat G (x ij, x lm ) log ( x x lm ). Tus from te order requirement of te interpolation function, we ave te first inequality. To prove te second inequality, we use te expression from (4.11) in [17]: G (x ij, x lm ) = g (x ij, x lm ) ŝ (x ij, x lm ) s (x ij, x lm ), (37) were ŝ(x ij, x lm ) is cosen suc tat G (x ij, x lm ) = 0 at te grid points on te boundary of te unit square using te metod of images. Tus ŝ(x ij, x lm ) is a combination of g (x ij, x lm ) at points outside of te unit square. From Teorem 3.1 in [18], we ave α g (x ij, x lm ) C/ ( x ij x lm 2 + ) α 1. Te same can be said for ŝ(x ij, x lm ). Te term s(x ij, x lm ) is cosen suc tat G (x ij, x lm ) = 0 at te grid points on te boundary of te unit square and s(x ij, x lm ) = 0 at all oter grid points. Tus s(x ij, x lm ) Ω is te trace of g (x ij, x lm ) ŝ(x ij, x lm ), see Remark on page 297 in [5]. Since te values of s(x ij, x lm ) Ω are from te discrete Laplacian, it can be smootly extended to te entire boundary Ω. Tus from Lemma in [9] and te maximum principle, we ave α s (x, x lm ) ( ) α 1 ( ) α 1 C α 1 C α 1 sup s = sup s, (38) dist(x, Ω) Ω dist(x, Ω) Ω were C is anoter constant. From Section 4.4 in [17], we know tat s(x ij, x lm ) = O(1) as long as dist( Ω, x lm ) = O(1). Tus as long as dist(x, Ω) O(1), we ave α s (x, x lm ) O(1). Tis completes te proof of te lemma. 11

12 Remark 3.5. Te interpolation function is not unique. Along te boundary Ω, from te requirement of te interpolation function, G I (x ij, x lm ) / n does exist and continuous. We ave, for example G I (x ij, x lm ) = G I (x 1j, x lm ) G I (x 0j, x lm ) + O() (39) x along te boundary x = 0. By similar procedure in proving te discrete Green s formula, we can get te second discrete Green s formula. Lemma 3.6. Let u(x, y) be te solution to (19) and G I (x, x lm) be an interpolation function of G (x ij, x lm ) tat satisfies te conditions listed in Section 3.2. If l, m 1 or n 1, ten we ave, u(x i, y j )G (x ij, x lm ) 2 = v(s)g I (X(s), x lm)ds + O(). ij Proof: Again, we sow te second discrete Green s formula by expanding te summation. After cancelation of interior terms, only boundary terms and a source are left. Tus, we get i,j=1 = u(x i, y j )G (x ij, x lm ) 2 = + j=1 + j=1 Ω j=1 u(x 0, y j ) u(x 1, y j ) G (x 1j, x lm ) u(x n, y j ) u(x, y j ) G (x,j, x lm ) + u(x i, y n ) u(x i, y ) G (x i,, x lm ) j=1 G (x n,j, x lm ) G (x,j, x lm ) u(x, y j ) G (x i,n, x lm ) G (x i,, x lm ) u(x i, y ) + u(x i, y 0 ) u(x i, y 1 ) G (x i1, x lm ) G (x 0j, x lm ) G (x 1j, x lm ) u(x 1, y j ) i,j=1 ( u n (x) G I (x, x lm ) G I (x, x ) lm) u(x) ds + u(x lm ) + O(). n G (x i0, x lm ) G (x i1, x lm ) u(x i, y 1 ) u(x i, y j ) G (x ij, x lm ) 2 On te oter and, by integrating bot sides of te partial differential equation (19) wit f = 0 and u 0 = 0, we get ( ) G I (x, x lm) udxdy = v(s)δ (x X(s)) (y Y (s)) ds G I (x, x lm)dxdy, Ω Ω or equivalently, ( u Ω n G I (x, x lm ) G I (x, x ) lm) u ds + u G I (x, x lm )dxdy = v(s) G I (X(s), x lm )ds. n Ω 12

13 Note tat u(x) G I (x, x lm)dxdy = Ω = R ij = R ij R ij R ij\r lm ( ) u(x ij ) + O() O()dxdy + Ω u(x) G I (x, x lm)dxdy + u(x) G I (x, x lm)dxdy R lm R lm u(x ij )O( 3 ) + u(x lm ) G I (x, x lm)dxdy + O() R lm = O() + u(x lm ) + O() = u(x lm ) + O(), ( ) u(x lm ) + O() G I (x, x lm ) dxdy were R ij is te square centered at x ij. Tis completes te proof. Lemma 3.7. Let Cij IB be te correction terms in te immersed boundary metod (23), v(s) C 1 be defined in (19), and G I (x, x lm) be an interpolation function of G (x ij, x lm ) tat satisfies te conditions listed in Section 3.2. Ten we ave te following estimate. Cij IB G (x ij, x lm ) 2 = v(s)g I (X(s), x lm )ds + O( log ). ij Proof: We denote s = max{ s k }. ij Cij IB G (x ij, x lm ) 2 = ij N b k=1 v k δ (x i X k ) δ (y j Y k ) s k G (x ij, x lm ) 2 N b = v k s k δ (x i x) δ (y j y) G I (x ij, x lm ) 2 k=1 ij N b ( ) = v k s k G I (X k, x lm ) + E k. k=1 From te expression (4.2) and Lemma 4.1 in [3], we know tat E k C α G I (ξ k, x lm ). α 1=1 Te summation N b k=1 v ( ) k s k G I (X k, x lm ) + E k is te composite trapezoidal rule for te line integral. We divide te summation into tree groups, one wit te summation of k tat X k x lm 2, one wit X k x lm 2, and te oter is for te rest of k s. Te contributions from te boundary points are split as alf and alf for eac group. For te first two groups, we ave ( ) v k s k G I (X k, x lm ) + E k = v(s)g I (X(s), x lm) ds + E 1, x lm X k x lm 2 13

14 were means te coefficients is alf at te two boundary points. For te part X k x lm 2, we ave E 1 C 2 1 s max k ( X k x lm 2 + ) 2 C 2 1 s ( ) 2 C, + due to te second order partial derivatives of G I (X(s), x lm) and te first order derivatives of E k. For te part X k x lm 2, te error estimate is a little bit tricky. From te error estimate of te trapezoidal rule in eac interval, we ave E 1 2 s 12 k s α 1=2 α G I (ξ k, x lm ) C 2 s s ( ξk x lm ) 2 + ) 2 k C 2 s s s dr ( ) 2 C 2 s dr s r + s r 2 C. For te last group, we ave ( ) v k s k G I (X k, x lm ) + E k X k x lm 2 = x lm v(s)g I (X(s), x lm ) ds + E 2 max v(s) log + E 2 s, from te estimate of G I (X(s), x lm) in (36). For te error term E 2, since te lengt of te integral is O(), we ave E 2 C 2 max X k 1 ( X k x lm 2 + ) 2 C 3 1 ( + ) 2 C. Tis completes te proof of te lemma. Now we are ready to prove te main result of te paper. Teorem 3.8. Let u(x, y) be te solution to (19) and U is te solution obtained from te immersed boundary metod (23) using a discrete delta function δ (x) for (19). Ten U is first order accurate wit a logaritm factor in te L norm, tat is, E ij C log, i, j = 1, 2,, n 1. (40) 14

15 Proof: Consider te error at a grid point E lm, if x lm is close to te interface, tat is, dist(, x lm ) W, we ave E lm = ( (A ) 1 T IB) lm = ( (A ) 1 T IB reg )lm + ( (A ) 1 T IB ) irr lm = O( 2 ) + ( (A ) 1 T IB irr lm ( ) = 2 T ij (A ) 1 1 e ij 2 = dis(x ij,) W dis(x ij,) W ) lm + O( 2 ) 2 ( u(x i, y j ) Cij IB ) G (x ij, x lm ) + O( 2 ) = ij 2 u(x i, y j )G (x ij, x lm ) ij 2 C IB ij G (x ij, x lm ) + O( 2 ) = ( v(s)g I (X(s), x lm)ds k v k G I (X k), x lm ) s k ) + O() = O( log ), after we apply Lemma 3.6 and Lemma 3.7. Note tat, in te expansion of te summation from dis(x ij, ) W to all interior grid points, we ave used te fact tat u(x i, y j ) = O( 2 ) and Cij IB = 0 wen dis(x ij, ) > W. If dist(, x lm ) > W, te proof above is still valid except tat we are not going to ave te singular integration. Tus, we do not need to ave te log factor. Tis means tat for IB metod, te larger errors often occur near or on te interface. 4 Conclusions and acknowledgments We give a convergence proof of te immersed boundary (IB) metod in te L norm. Te key of te proof is to establis a connection between te discrete Green function and a continuous one wit te same boundary conditions. We sow tat te IB metod is indeed first order accurate wit a log factor if a reasonable discrete delta function is used. Te conclusion sould be applicable to oter linear boundary conditions in addition to Diriclet type as long as te metod of images apply. Te autor was partially supported by te US ARO grants MA, te AFSOR grant FA , te US NSF grant DMS , and te NIH grant Te autor is grateful for many valuable suggestions and comments from Dr. J. Tomas Beale of Duke University and Dr. Kazufumi Ito of Nort Carolina State University. 15

16 References [1] J. Adams, P. Swarztrauber, and R. Sweet. Fispack: Efficient Fortran subprograms for te solution of separable elliptic partial differential equations. ttp:// [2] J. T. Beale and A. T. Layton. On te accuracy of finite difference metods for elliptic problems wit interfaces. Commun. Appl. Mat. Comput. Sci., 1:91 119, [3] R. P. Beyer and R. J. LeVeque. Analysis of a one-dimensional model for te immersed boundary metod. SIAM J. Numer. Anal., 29: , [4] K.-Y. Cen, K.-A. Feng, Y. Kim, and M.-C. Lai. A note on pressure accuracy in immersed boundary metod for stokes flow. J. Comput. Pys., 230: , [5] L. C. Evans. Partial Differential Equations. AMS, [6] K. Gürlebeck and A. Hommel. On finite difference potentials and teir applications in a discrete function teory. Mat. Met. Appl. Sci., 25: , [7] W. Hackbusc. Elliptic Differential Equations: Teory and Numerical Treatment. Springer- Verlag, [8] H. Hasimoto. On te periodic fundamental solutions of te Stokes equations and teir application to viscous flow past a cubic array of speres. J. Fluid Mec., 5: , [9] N. V. Krylov. Lectures on Elliptic and Parabolic Equations in Holder Spaces. AMS, [10] G. F. Lawler. Intersections of random walks. Birkauser, [11] R. J. LeVeque and Z. Li. Te immersed interface metod for elliptic equations wit discontinuous coefficients and singular sources. SIAM J. Numer. Anal., 31: , [12] Z. Li. Te Immersed Interface Metod A Numerical Approac for Partial Differential Equations wit Interfaces. PD tesis, University of Wasington, [13] Z. Li and K. Ito. Te Immersed Interface Metod Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. SIAM Frontier Series in Applied matematics, FR33, [14] Y. Mori. Convergence proof of te velocity field for a Stokes flow immersed boundary metod. Comm. Pure Appl. Mat., 61:2008, [15] C. S. Peskin. Numerical analysis of blood flow in te eart. J. Comput. Pys., 25: , [16] C. S. Peskin and D. M. McQueen. A general metod for te computer simulation of biological systems interacting wit fluids. Symposia of te Society for Experimental Biology, 49:265, [17] V. Rutka. Immersed Interface metods for elliptic boundary value problems. PD tesis, University of Kaiserslautern,

17 [18] V. Tomée. Discrete interior scauder estimates for elliptic difference operators. SIAM J. Numer. Anal., 5: , [19] A.-K. Tornberg and B. Engquist. Numerical approximations of singular source terms in differential equations. J. Comput. Pys., 200: , [20] A.-K. Tornberg, B. Engquist, and R. Tsai. Discretization of Dirac delta functions in level set metods. J. Comput. Pys., 207:28 51, [21] S. Zang. On te full C1-Qk finite element spaces on rectangles and cuboids. Adv. Appl. Mat. Mec., 2: , A Te construction of an interpolation function of te discrete Green function We construct an interpolation function from te grid function G I (x ij, x lm ) to of te discrete Green function G I (x, x lm) tat satisfies te conditions listed on page 10. Let φ(x, y) be te function tat satisfies te following conditions: G I (x, x lm ) = δ (x, y), lm δ (x, y)dxdy = 1 2. (41) G I (x, x lm) takes values of G I (x ij, x lm ) at four corners and te center. In oter words, we coose te source term and te boundary condition so tat te above two conditions are satisfied. Ten, we construct te interpolation function G I (x, x lm) on oter squares from te values of G I (x, x lm) using Q k (K) C 1 (Ω), were Q k is defined as te finite element space k Q k (K) = v(x, y), on eac R îĵ : v(x, y) = β ij x i y j, v(x, y) C 1 (Ω) (42) i=0,j=0 over a quadrilateral mes K, Rîĵ is a square of by. In te paper [21], te autor as proposed a system way to construct C 1 -Q k finite element spaces on quadrilaterals meses. We ave implemented and tested te interpolation function G I (x, x lm) of Q 5 (K) C 1 (Ω), wic is tird order accurate (k = 3). As usual, we just need to construct a sape function over a unit square. Te total degree of freedom of Q 5 (K) C 1 (Ω) is 36. In our construction, we specify te values of v, v x, v y, v xx, v xy, v yy, wic imposes 24 constraints. To keep te continuity of te solution and first order partial derivatives, for example, along te side x = 0, we impose te coefficients of xy 5 and xy 4 to be zero. Tat is, tere are two constraints along eac side, wic adds additional 8 17

18 constraints. Using te undetermined coefficient metod for te 36 constraint wit te 32 conditions, we ave a system of equations wit 36 unknowns and 32 equations (constraints). Te condition number of te coefficient matrix, te ration of te largest and smallest non-zero singular values is , wic indicates te matrix as full row rank and te system of equations as infinite number of solutions. We suggest to coose te SVD solution as te interpolation function. To see tat te interpolation function is in C 1, we use te side x = 0 as an example. Along tis side, te function is a fift polynomial, wic is uniquely determined by its values, te first and second order derivatives at two points. Tus te interpolation function and its tangential derivative are continuous along tis side. Te normal derivative is in general still a fift polynomial of y along x = 0. After we impose te constraints tat its coefficients of xy 5 and xy 4 to be zero, it becomes a cubic polynomial wic is uniquely determined by its values (v x ) and tangential derivatives (v xy ) at two points. In figure 3, we sow a plot of suc an interpolation function in two neigboring square wit function value at one corner be a unit wile oters and all te derivatives are zero Figure 3: A plot of te sape function in Q 5 (K) C 1 (Ω) at two neigboring squares. B A numerical example We use a numerical example to sow tat te immersed boundary metod is indeed first order accurate and its local truncation error is of order O(1/). Te differential equation is u(x, y) = 2δ(x X(s))δ(y Y (s))ds, 1 < x, y < 1 (43) 18

19 were te interface is te circle r = 1/2, were r = x 2 + y 2. Te function u(x, y) = { 1 if r log(2r) if r > 1 2 (44) satisfies te PDE. Tis example is taken from [11]. We use te IB metod to solve te PDE wit te Diriclet boundary condition given by (44). In Table 1, we list te results of a grid refinement analysis. In Table 1, te number N is te number of grid lines in te x and y directions. Te interface r = 1/2 is discretized by X k = 0.5 cos(2kπ/n) Y k = 0.5 sin(2kπ/n), k = 0, 1,, N 1. N E cos order cos T cos E at order at T at Table 1: A grid refinement analysis of te IB metod using discrete cosine and at delta functions. In te second and fift columns of Table 1, we sow te L errors of te computed solution of te IB metod using te discrete cosine and at delta functions, respectively. Te tird and sixt columns, we sow te approximate convergence order using te two consecutive errors. We see clearly first order convergence. In te fourt and sevent columns, we sow te local truncation errors in te L norm. We see tat te local truncation error increase in te order of O(1/). 19