Centrum voor Wiskunde en Informatica

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1 Centrum voor Wiskunde en Informatica MAS Modelling, Analsis and Simulation Modelling, Analsis and Simulation Discontinuous Galerkin discretisation with embedded boundar conditions P.W. Hemker, W. Hoffmann, M.H. van Raalte REPORT MAS-R3 MARCH 3, 23

2 CWI is the National Research Institute for Mathematics and Computer Science. It is sponsored b the Netherlands Organization for Scientific Research (NWO). CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronms. Probabilit, Networks and Algorithms (PNA) Software Engineering (SEN) Modelling, Analsis and Simulation (MAS) Information Sstems (INS) Copright 2, Stichting Centrum voor Wiskunde en Informatica P.O. Bo 9479, 9 GB Amsterdam (NL) Kruislaan 43, 98 SJ Amsterdam (NL) Telephone Telefa ISSN

3 Discontinuous Galerkin Discretisation with Embedded Boundar Conditions P.W. Hemker,, W. Hoffmann and M.H. van Raalte, CWI, P.O. Bo 9479, 9 GB Amsterdam, The Netherlands KdV Institute for Mathematics, Universit of Amsterdam Plantage Muidergracht 24, 8 TV Amsterdam, The Netherlands ABSTRACT The purpose of this paper is to introduce discretisation methods of discontinuous Galerkin tpe for solving second order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundar. The methods aim at high-order accurac and the difficult arises since the regular grid cannot follow the curved boundar. Starting with the Lagrange multiplier formulation for the boundar conditions, we derive variational forms for the discretisation of 2-D elliptic problems with embedded Dirichlet boundar conditions. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DG-discretisation is adapted in the cells with embedded boundaries. We give eamples of approimation with tensor products of cubic polnomials. As an illustration, we solve a convection dominated boundar value problem on a comple domain. Although, of course, it is impossible to accuratel represent a boundar laer with a comple structure b means of a cubic polnomial, the boundar condition treatment appears quite effective in handling such comple situations. 2 Mathematics Subject Classification: 65N5; 65N99 Kewords and Phrases: DG discretisation, structured grid, irregular boundar, embedded boundar Note: This work was carried out under project MAS2. Computational Fluid Dnamics.. Introduction The purpose of this paper is to introduce methods of discontinuous Galerkin tpe for solving second order elliptic PDEs on a structured, regular rectangular grid while the problem is defined on a curved boundar. The methods aim at high-order accurac and the difficult arises because the regular grid cannot follow the curved boundar. Earlier, several techniques were proposed to handle boundar conditions on irregular, curvilinear boundaries. The most convenient certainl is the FEM, where elements near the boundar are adapted to the shape of the boundar curve. Generall, this results in an unstructured grid. This relativel straightforward technique can be applied up to arbitraril high-order of accurac and delivers good results.

4 . Introduction 2 In contrast, finite difference methods are usuall applied on regular grids. Here, curved boundaries are treated b locall adapted finite differences as, e.g., Shortle-Weller approimation [4, Sect.4.8]. Generall, such discretisations are not used for higher orders of accurac. A more recent technique for treatment of comple boundaries on orthogonal grids, in two or three dimensions, is the Embedded Curved Boundar (ECB) method. Here usuall in the contet of the discretisation of conservation laws piecewise linear segments are embedded in the grid to represent the boundar. The method is also used, e.g., for solutions across interfaces [7, 8]. In man cases the ECB method shows clear advantages compared to the traditional stair-step method [] but no higher-order accurac than order two can be epected. Higher order ma be obtained b Immersed Boundar Methods (IBM) [, 2, 3, 6], e.g., in pseudo-spectral codes [3], where the presence of a boundar within the computational domain is simulated b specifing a bod force term, without altering the computational grid. This technique is ver fleible as it allows for bodies and interfaces of almost arbitrar shape. The method is quite popular in situations with interfaces and rather comple geometries [9] and e.g., elastic boundaries [5]. Usuall the method is applied so as to maintain second order accurac (first order near the boundaries). However, fourth order convergence rates are reported in [2], where the same methodolog is used with PDEs for thin fleible membranes in an incompressible fluid domain. In contrast with the above methods, we take the Lagrange multiplier formulation of the boundar conditions as a starting point, in the same manner as used in [4] or in the derivation of the discontinuous Galerkin discretisation. Within the framework of structured, regular rectangular grids we introduce the treatment of curved boundaries in full agreement with the principles that lead to the discontinuous Galerkin method. We appl high-order DGdiscretisation in the interior and adapt the method in the cells with embedded boundaries. The order of approimation of the boundar condition corresponds with the accurac of the DG-method. In the present paper we give eamples of approimation with tensor products of cubic polnomials. In [6] we eplained wh the treatment of this cubic polnomial case is the basis for higherorder approimation. In DG discretisation, information echange over the interior cell boundaries is restricted to function values and normal flues. At the endpoints of an interval, function values and flues are determined b four independent parameters, that correspond with the four degrees of freedom in the cubic polnomial approimation on a cell. Higher-order approimation can be achieved b additional bubble functions with vanishing values and derivatives at the cell boundar. In the multi-dimensional case, on a structured rectangular grid, the same principle holds with tensor-products for approimation. For the treatment of the embedded boundar conditions, we give in Section 2 of this paper an eposition of the weak forms used for the different discretisation alternatives. In Section 3 we start with simple eperiments in one and two dimensions to see the differences between the various methods. In section 4 we identif the discrete function spaces in which the approimate solution is found. In the last section we solve a convection-dominated equation on an irregular domain, partitioned into onl two cells. We show how well, on this mesh, a comple problem can be solved with a piecewise cubic approimation.

5 2. Weak forms for the Poisson equation 3 2. Weak forms for the Poisson equation 2. The Lagrange multiplier form for the embedded boundar problem To appl DG-methods for structured rectangular grids on complicated domains, we are interested in solving an elliptic second order problem Lu = f on a fictitious open domain Ω, which is larger than the open domain Ω on which the elliptic BVP is originall defined. The solution u on Ω is determined b the Dirichlet boundar condition u = u on Ω, the boundar of Ω, and we want to discretize the problem on a fictitious domain Ω Ω. For this purpose we assume that the solution u on Ω allows a sufficientl smooth etension, u, defined on Ω, solving Lu = f. Of course, this ecludes certain tpes of singularities near the boundar. For sake of simplicit, in this initial treatment we assume Ω to be the unit cube and we consider the Poisson equation with an embedded Dirichlet boundar condition as follows: let Ω be the open unit cube, with boundar Ω, which consists of two non-overlapping open sub-domains, Ω and Ω, such that Ω =Ω Ω, and Ω Ω =, (2.) where Ω is the fictitious part of the domain Ω. We now consider the boundar value problem consisting of the Poisson equation defined on the whole of Ω and Dirichlet boundar conditions on Ω, the boundar of Ω: Lu u = f on Ω, and u = u on Γ D = Ω, (2.2) under the assumption that the solution u on Ω, has a sufficientl smooth continuation to Ω, satisfing the Poisson equation on the whole of Ω. Ω Ω Ω Ω Ω Figure : The domain of interest, Ω and the fictitious part, Ω, make the domain Ω =Ω Ω. To arrive at the corresponding weak formulation of the Poisson equation with embedded Dirichlet boundar condition, we multipl the left- and right-hand side of (2.2) with a sufficientl smooth function v, and integrate over the domain Ω, to get: find u H ( Ω) such that ( u, v) Ω n u, v Ω =(f,v) Ω v H ( Ω), (2.3) under the constraint that u = u on Ω. B the Lagrange multiplier theorem, the following formulation is equivalent to (2.3): find u H ( Ω) and p H /2 ( Ω) such that ( u, v) Ω n u, v Ω + p, v Ω =(f,v) Ω v H ( Ω), (2.4) q, u Ω = q, u Ω q H /2 ( Ω).

6 2. Weak forms for the Poisson equation 4 We call this the Lagrange multiplier form of the embedded boundar problem. We see that, if u satisfies the Poisson equation (2.2) and the embedded Dirichlet boundar condition, the Lagrange multiplier p in (2.4) vanishes. 2.2 The weak form for boundaries along gridlines In the classical case that Ω =, we can combine the boundar terms in (2.4), in order to obtain ( u, v) Ω p, v Ω =(f,v) Ω q, u Ω = q, u Ω v H ( Ω), q H /2 ( Ω), with p = n u p on Ω = Ω. This leads to a hbrid form of (2.2) with Dirichlet BCs: find u H ( Ω) and p H /2 ( Ω) such that ( u, v) Ω p, v Ω q, u Ω =(f,v) Ω q, u Ω v H ( Ω),q H /2 ( Ω). (2.5) When u satisfies (2.2) we have p = n u, the normal flu at the boundar Ω. Substituting this value for p, and replacing similarl the weighting function q b q = σn v, with σ =orσ =, leads to the weak form used in DG-methods (viz., Baumann s and the smmetric DG-method respectivel). Other DG-methods (viz., IPG, NIPG) are obtained b taking q = σn v µv with parameters σ and µ. Thus, our DG weak form reads: find u H ( Ω) such that ( u, v) Ω n u, v Ω + σ n v, u Ω = (2.6) =(f,v) Ω + σ n v, u Ω v H ( Ω). 2.3 The hbrid and the DG-form for the embedded boundar problem Not onl the Lagrange multiplier form (2.4) can be used for the embedded boundar problem, we can also appl (2.5) or (2.6). In the case Ω the form (2.5) reads: find u H ( Ω) and p H /2 ( Ω) such that ( u, p, v v) Ω q, u Ω Ω q, u =(f,v) Ω Ω, (2.7) v H ( Ω),q H /2 ( Ω). which we call the hbrid form of the interior boundar problem. In the case Ω, equation (2.6) is written: find u H ( Ω) such that ( u, v) Ω n u, v Ω + σ n v, u Ω = (2.8) =(f,v) Ω + σ n v, u Ω v H ( Ω), which we call the DG-form (the Baumann-Oden weak form if σ = or the smmetric form if σ = ) of the interior boundar problem. Notice that this smmetric weak form is not smmetric anmore if Ω Ω.

7 3. Numerical eperiments in one and two dimensions 5 3. Numerical eperiments in one and two dimensions 3. Numerical eperiments on one-dimensional problems To see the difference in practice, we first stud the three weak forms (2.4), (2.7) and (2.8) for a simple one-dimensional problem. On the unit interval Ω =(, ) we consider the Poisson equation with homogeneous Dirichlet boundar conditions: d2 u d 2 = f, on Ω, with u(d) =, u() =, (3.) where d [, ) and Ω = (d, ). To discretize this problem we take for test and trial spaces the (p + )-dimensional space S h ( Ω) = P p ( Ω) H ( Ω), i.e., the space of polnomials of degree p. We write for the approimate solution u h = c i φ i (), φ i () S h ( Ω). i p Further, we provide the boundar spaces Q h ( Ω) H /2 ( Ω) and Q h ( Ω) H /2 ( Ω), with the trace of polnomials on the boundar, hence Q h ( Ω) = {ψ () =( ) =(,), ψ () = =(,) )} and Q h ( Ω) = {ψ () = d =(d,), ψ () = d d =(d,))}, d [, ). Then we write for the approimation of the Lagrange multiplier: p h = a i ψ i () =,d,. i Because of the -D character of this eample, boundar values are parameterized b onl two values for both Ω and Ω. Given the approimating spaces, the three forms (2.4), (2.7) and (2.8) become: (i) In case of the Lagrange multiplier formulation: find u h S h ( Ω), p h Q h ( Ω) such that u h v h d [ u h ()v h() u h ()v h() ] +[p h ()v h () p h (d)v h (d)] (3.2) +[q h ()u h () q h (d)u h (d)] = v h fd, v h S h ( Ω), q h Q h ( Ω). (ii) in case of the hbrid form: find u h S h ( Ω), p h Q h ( Ω) such that u h v h d [p h()v h () p h ()v h ()] [q h ()u h () q h (d)u h (d)] (3.3) = v h fd, v h S h ( Ω), q h Q h ( Ω). (iii) whereas the DG-formulation reduces to: find u h S h ( Ω) such that u h v h d [u h ()v h() u h ()v h()] + σ [v h ()u h() v h (d)u h(d)] (3.4) = v hfd, v h S h ( Ω).

8 3. Numerical eperiments in one and two dimensions Figure 2: The solution u() = Smmetric-Baumann- method, σ = ±. computed with the hbrid-, Lagrange- and As a first eperiment we check if the three discrete forms (3.2), (3.3) and (3.4) can solve for the eact solution, when we choose f() = in (3.) and d =/2, and if we take S h ( Ω) = P 3 ( Ω). The result is shown in Figure 2. It appears that all three formulations compute the eact solution. The computed Lagrange-multipliers for the hbrid- and Lagrange-formulation are shown in Table. We see that, in case of the hbrid-formulation, the Lagrange multipliers correspond with the flues at the boundaries, i.e., p h () = du dn () and p h() = du dn (), whereas for the Lagrange-formulation, the Lagrange multipliers vanish. Lagrange method (3.2) p h (d) = p h () = hbrid method (3.3) p h () = 5/25 p h () = 7/24 Table : The values of the Lagrange multipliers of hbrid- and Lagrange- method for the solution as in Figure 2. Net we check if we can solve (2.2) for an arbitrar location d [, ) of the interior Dirichlet boundar condition. Now we see that the dependencies on d and σ differ for the three methods. In case of the smmetric-/baumann-oden method we have to solve a full (p +) (p + ) linear sstem L σ,d u h = f h, where the matri depends on both the method parameter σ and the interior boundar location d. In contrast, if we consider the coefficients of the linear sstem arising from the hbrid- and the Lagrange method we observe the following block-partitioning: L d u h = ( A B C ) u h = f h, where, for the Lagrange method, A = u h v h d [u h (, )v h(, )] is the (p +) (p +) leading submatri, and B = [p h (d, )v h (d, )] and C = [q h (d, )u h (d, )] have respectivel dimensions (p +) 2 and 2 (p + ). The dependenc on d is reflected in the elements of B and C.

9 3. Numerical eperiments in one and two dimensions 7 On the other hand, in case of the hbrid-method, we have a (p+) (p+) leading submatri A = u h v h d. Now the (p +) 2 submatri B = [p h (, )v h (, )] is independent of d. The dependenc on d is onl reflected in the 2 (p +)matric = [q h (d, )u h (d, )]. So we check if there are locations d [, ) in which an of the three methods ma become singular. The results are shown in Table 2. We see that both the Lagrange- and the smmetric-/baumann-method have interior boundar locations where the methods become singular. The number of points where a singularit appears increases with the polnomial degree. The hbrid-method, however, shows no singular points. This motivates us to continue mainl with the hbrid method for the two-dimensional numerical eperiments. The Lagrange method p =2 /3 p =3 2/5 / 6 2/5+/ 6 p = The smmetric- / Baumann-method p =2 p =3 2/5 p =4 3/7 /7 2 3/7+/7 2 Table 2: Values of d for which the discrete sstem becomes singular. The discretisations are made for S h ( Ω) = P p (, ), p =2, 3, Numerical eperiments for the hbrid-method on two-dimensional problems Having studied the one-dimensional discretisation for the various weak formulations with an embedded Dirichlet boundar condition, we now consider the two-dimensional Poisson equation on the unit square Ω as in (2.2) with an embedded Dirichlet boundar condition on a line parallel to a diagonal. For this line we use the following parametrisation (See Figure 3 ): { (s) = d/ 2+s, (s) = d/ 2 s, with { s <d/ 2 if d / 2, s < d/ 2 if / 2 <d< 2. (3.5) Approimation with piecewise quadratics. To discretize the hbrid formulation, we first introduce the quadratic polnomial basis on the unit interval P 2 ([, ]) = Span{ t, t, t( t)}. (3.6) We provide the test and trial function spaces with the 9 dimensional subspace S h ( Ω) = P 2 2 ( Ω) = P 2 () P 2 () H ( Ω), i.e., the tensor product set of polnomials of degree 2 in the two coordinate directions. Since we know that the Lagrange multiplier of the hbrid-method corresponds with the flu p = n u on the boundar Ω, we choose to discretize the Lagrange multiplier as p h = n ψ (, ) Ω + n ψ (, ) Ω,withψ P 2 2 ( Ω) which defines the polnomial subspace Q h ( Ω) H /2 ( Ω) and also Q h ( Ω) H /2 ( Ω)

10 3. Numerical eperiments in one and two dimensions 8 d d Figure 3: The domain Ω and its parametrisation b p h Ω = n ψ (, ) Ω + n ψ (, ) Ω. Then the discrete formulation of the hbrid form is: find u h S h ( Ω), p h Q h ( Ω) such that u h, v h Sh ( Ω) p h,v h Qh ( Ω) = f,v h Sh ( Ω), v h S h ( Ω), (3.7) q h,u h Qh ( Ω) = q, u Qh ( Ω), q Q h( Ω), where the approimations are given b (9 degrees of freedom describe the polnomial in the interior) u h (, ) = c i φ i (, ), φ i S h ( Ω), (, ) Ω, (3.8) i 8 and (notice that 8 degrees of freedom describe the quadratic polnomials at the 4 boundaries) p h (, ) = [ a i n ψ,i (, ) Ω + n ψ,i (, ) Ω], (3.9) i 7 ψ,i Ω,ψ,i Ω Q h ( Ω), (, ) Ω. Theresultisa7 7 linear sstem depending on the diagonal distance of the embedded Dirichlet boundar to the origin. It is obvious that all methods will become ill-conditioned for values of d close to 2, when the region Ω vanishes. In order to see how the singularit develops for the hbrid method (2.7), we plot the 7 singular values as function of the diagonal distance d. The result is shown in Figure 4. We see that, as in the one-dimensional eperiment, also for this eperiment, there are no values of d for which the discretisation matri becomes singular. Furthermore the method is not ill-conditioned for values of d even larger than one. Approimation with piecewise cubics. Net we want to stud the stabilit of a higher order discretisation of the hbrid method (2.7), for the same two-dimensional model problem. For that we consider on the unit interval the cubic polnomial basis P 3 ([, ]) = { t, t, t( t) 2,t 2 ( t) }, (3.) and we choose for the test- and trial function spaces the 6-dimensional subspace S h ( Ω) = P 3 3 ( Ω) = P 3 () P 3 () H ( Ω), i.e., the tensor product polnomials of degree less

11 4. Weak forms for embedded boundar conditions 9 than four in the two coordinate directions. We choose the polnomial subspaces Q h ( Ω) = γ Ω (S h ( Ω)) H /2 ( Ω) and Q h ( Ω) = γ Ω(S h( Ω)) H /2 ( Ω). The choice of the basis functions in Q h ( Ω) will be eplained in the net section, where we stud the general case with a curved boundar. As eplained in Section 4, using (3.7) and (3.8) we obtain a sstem depending on the diagonal distance d of the interior Dirichlet boundar to the origin. For this hbrid discretisation, the 28 singular values as function of d are shown in Figure 5. Generall, we observe the same behavior as for the quadratic polnomials..e2...e.e 2.e 3 e 5 e 6 e 7 e 8 e 9 e e e 2 e 3 e 4 e 5 e Figure 4: Singular values σ i, i 7 as function of the diagonal distance d< 2for the third order discretisation of the hbrid method..e e.e 2.e 3 e 5 e 6 e 7 e 8 e 9 e e e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 2 e 2 e 22 e 23 e 24 e 25 e 26 e 27 e 28 e 29 e 3 e 3 e 32 Figure 5: Singular values σ i, i 28 as function of the diagonal distance d< 2for the fourth order discretisation of the hbrid method. 4. Weak forms for embedded boundar conditions 4. The boundar condition on a curved embedded boundar In this section we stud the stabilit and the accurac of a fourth order hbrid discretisation of the Poisson equation on the unit square, with a part of the circular boundar embedded.

12 4. Weak forms for embedded boundar conditions So, we solve the equation u = f on Ω, with u = u on Γ D = Ω, (4.) on the unit square from which a circle sector has been removed: i.e., Ω the unit square and Γ int Ω, with Ω Ω, is the circular curve Γ int = { (, ) = R 2 <,, }, (4.2) and the fictitious part is Ω = { (, ) >, >, <R 2}. (4.3) The corresponding discrete hbrid formulation reads: find u h S h ( Ω), χ h Q h ( Ω) such that ( u h, v h γ ) Ω Ω (χ h ),γ Ω (v h ) =(f,v Ω h) Ω, v h S h ( Ω), (4.4) γ Ω (q h ),γ Ω (u h ) Ω = γ Ω (q h ),u q h Q h ( Ω), Ω, where S h ( Ω) H ( Ω) and Q h ( Ω) H ( Ω) are proper finite dimensional polnomial subspaces, and γ Ω, γω and γ Ω, γ Ω are the usual trace operators on Ω and Ω respectivel. To provide these subspaces with a basis, we choose cubic polnomials and consider the following polnomial basis on the unit interval: φ = t, φ 2 =( t) 2 t, φ 4 = t, φ 3 =( t)t 2. (4.5) We recognize that φ (t) andφ 4 (t) are associated with function values at t =, respectivel, while φ 2 (t) andφ 3 (t) can be associated with corrections for the derivatives at t =,. These facts help us to understand the structure behind the different polnomial subspaces that are constructed below. First we choose for the test and trial function spaces the 6-dimensional subspace, i.e., S h ( Ω) = P 3 3 ( Ω) = P 3 () P 3 () H ( Ω), the usual tensor product of polnomials of degree less than four in the two coordinate directions. Hence, on the unit square Ω weget the approimation u h S h ( Ω), u h = c i,j φ i ()φ j (). (4.6) i,j 4 Net we consider the usual trace operators, γ Ω : H ( Ω) H /2 ( Ω) and γ Ω : H ( Ω) H /2 ( Ω) applied to the boundar of Ω and Ω respectivel, and similarl γ Ω : H ( Ω) H /2 ( Ω) and γ Ω : H ( Ω) H /2 ( Ω) the traces for the normal derivatives. We see that the approimating space of tensor product cubics, S h ( Ω) H ( Ω), is a 6-dimensional subspace. The trace of this space on Ω, the space γ Ω (S h ( Ω)), however, is 2-dimensional, because the trace consists of independent cubics on the four edges, related b four continuit conditions at the vertices. B the choice of the polnomial basis (4.5), in S h ( Ω) a basis for

13 4. Weak forms for embedded boundar conditions γ Ω (S h ( Ω)) can readil be found as a subset of the tensor product of basis functions (4.5), b splitting S h ( Ω) in two linearl independent subspaces: with and S h ( Ω) = Q h ( Ω) K h ( Ω), K h ( Ω) = ker(γ Ω ) S h ( Ω) = Span ( φ i ()φ j () i, j =2, 3 ), (4.7) Q h ( Ω) = Span (φ ()φ j (), φ 4 ()φ j (), φ i ()φ (), φ i ()φ 4 (); i, j =, 2, 3, 4). (4.8) For the approimating space for γ Ω (H ( Ω)) we take Q h ( Ω) := γ Ω (S h ( Ω)) = γ Ω ( Q h ( Ω)) H /2 ( Ω). Similarl we introduce the approimation space for the traces on Ω as Q h ( Ω) := γω ( Q h ( Ω)) H /2 ( Ω). On the other hand, for the approimation of the trace of the normal derivatives we split the space S h ( Ω) as with S h ( Ω) = Q h ( Ω) K h ( Ω) K h ( Ω) = ker(γ Ω ) S h ( Ω) = Span ( ψ i ()ψ j () i, j =, 4 ) with ψ k = φ k +( ) k (φ 3 φ 2 ), and Q h ( Ω) = Span (φ 2 ()φ j (), φ 3 ()φ j (), φ i ()φ 2 (), φ i ()φ 3 (), i,j=, 2, 3, 4). We see that Q h ( Ω) is 2-dimensional and K h ( Ω) is 4-dimensional. The normal derivatives on the four edges of Ω are all approimated b cubic polnomials that are related b the condition that at the vertices ( u h )= ( u h ). So we find the approimating space for the normal derivatives at the boundar of Ω, viz., Q h ( Ω) := γ Ω (S h ( Ω)) = γ Ω ( Q h ( Ω)) H /2 ( Ω) and at the boundar of Ω as Q h ( Ω) := γ Ω ( Q h ( Ω)) H /2 ( Ω). Considering the Lagrange multiplier function p H /2 ( Ω) in (2.7), we know that, if u satisfies Poisson s equation (4.) and also the Dirichlet boundar condition, the Lagrange multiplier p on Ω represents the normal flu n u at the the boundar Ω, i.e., p = n u. So, in the discrete hbrid formulation (4.4), we write for the Lagrange multiplier p h = n χ h

14 4. Weak forms for embedded boundar conditions 2 on Ω, where n is the unit outward normal vector and χ h Q h ( Ω) is the master flu function given b χ h = a i,j φ i ()φ j (), with a i,j =, i,j =, 4. (4.9) i,j 4 So we recognize the discrete hbrid formulation (4.4): find u h S h ( Ω), χ h Q h ( Ω) such that ( u h, v h γ ) Ω Ω (χ h ),γ Ω (v h ) =(f,v Ω h) Ω, v h S h ( Ω), (4.) γ Ω (q h ),γ Ω (u h ) Ω = γ Ω (q h ),u q h Q h ( Ω), Ω, as a (6 + 2) (6 + 2) linear sstem. To stud the stabilit of this hbrid formulation we plot the singular values of the discrete sstem as function of the circle radius, <R 2. The result is shown in Figure 6. In this figure we see 28 singular values as a function of the circle radius, R. The discrete formulation is sufficientl stable up to circle radii of R.. In that case more than 8% of the total domain Ω consists of the fictitious domain Ω. A reason for the cusps in the figure near R =.4 andr =.9 is unknown. Net, we check how the cubic approimation will solve for the eact solution b taking in (4.) the right-hand side and the boundar conditions such that the solution is given b u = The solution and the error for two possible domains (R =2/5 andr =4/5) are shown in the Figures 7 and 8. We see that the hbrid formulation finds the eact solution on the domain Ω, ecept for rounding errors which correspond with the condition of the linear sstem. To check the approimation behavior of the method we repeat the eperiment for the solution u(, ) =e + in (4.). The solution and the error for both domains (R =2/5 and R =4/5) are shown in the Figures 9 and....e.e 2.e 3 e 5 e 6 e 7 e 8 e 9 e e e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 2 e Figure 6: The singular values as function of the embedded circle bow radius for the fourth order hbrid discretisation. On the fictitious part of the domain the solution and the error are set equal to zero.

15 4. Weak forms for embedded boundar conditions 3 Error on domain of interest e 8 8e 9 6e 9 4e 9 2e 9 2e 9 4e 9 6e Figure 7: The solution u = and the error on the domain Ω of the fourth order hbrid discretisation (R =2/5). Error on domain of interest e 8 2 5e e 9 e 8.5e Figure 8: The solution u = and the error on the domain Ω of the fourth order hbrid discretisation (R =4/5). 4.2 The combination of the hbrid and the discontinuous Galerkin formulation In the previous sections we have seen that the hbrid form is stable on a cell with an embedded Dirichlet boundar condition, whereas the discontinuous Galerkin method is not alwas stable. On the other hand the discontinuous Galerkin method is cheaper, because the Lagrange multiplier has been eliminated and hence less degrees of freedom are involved. So, to cut down on computational costs, if we consider a large regular rectangular grid on which locall there eist cells with embedded Dirichlet boundar conditions, it is natural to treat these cells with the hbrid method for stabilit, while the normal rectangular cells are treated with a DG-Galerkin discretisation. To stud such a method, we consider two adjacent rectangular cells Ω and Ω 2,where onl Ω has an embedded Dirichlet boundar condition. The cells have a common interface Γ,2. Because cell Ω has an embedded boundar condition, we treat this cell with a hbrid

16 4. Weak forms for embedded boundar conditions 4 Error on domain of interest Figure 9: The solution u = e + and the error on the domain Ω of the fourth order hbrid discretisation (R =2/5). Error on domain of interest Figure : The solution u = e + and the error on the domain Ω of the fourth order hbrid discretisation (R =4/5). discretisation. Cell Ω 2 is discretized b DG discretisation. So, on cell Ω we have u h u h d (n,2 χ h )v h ds (n χ h )v h ds (4.) Ω Γ,2 Ω \Γ,2 (n,2 q h )u h ds (n q h )u h ds Γ,2 Ω \Γ,2 = fv h d n q h u ds, Ω Ω \Γ,2 where n is the unit normal on the interface Γ,2 pointing from cell Ω towards cell Ω 2.Onthe

17 4. Weak forms for embedded boundar conditions 5 other hand, on cell Ω 2 we consider (for simplicit) the smmetric DG discretisation. Hence u h u h d (n u h )v h ds (n u h )v h ds (4.2) Ω 2 Γ,2 Ω 2 \Γ,2 (n v h )u h ds (n v h )u h ds Γ 2, Ω 2 \Γ,2 = fvd (n v h )u ds, Ω 2 Ω 2 \Γ,2 NowwehavetocouplethetwocellsattheinterfaceΓ,2. Therefore, we have to satisf the localit, consistenc and conservation conditions as discussed in []. To meet these conditions we define average flues across the interface b uh = 2 ( χ h + u Ω h Ω2 )and vh = 2 ( q h + v Ω h Ω2 ), (4.3) and jumps b [u h ]=u h Ω n,2 + u h Ω 2 n 2,. (4.4) Then combining (4.) and (4.2), together with the flu and jump relations, we arrive at the form Ω Ω 2 u h v h d = fv h d Ω Ω 2 Γ,2 uh [v h ] ds (n χ h )v h ds Ω \Γ,2 (n q h )u h ds Ω \Γ,2 (n q h )u ds Ω \Γ,2 Γ,2 vh [u h ] ds (4.5) (n u h )v h ds Ω 2 \Γ,2 (n v h )u h ds Ω 2 \Γ,2 (n v h )u ds. Ω 2 \Γ,2 This weak form can immediatel be used for discretisation as described above. The solution and the error of such a combined discretisation with cubic polnomials is shown in Figure. 4.3 An embedded boundar for the convection equation Having studied the discretisation of the Poisson equation, we now consider the convection equation with an interior Dirichlet boundar condition b u = f in Ω, u = u on Ω in, (4.6) where b is a constant vector denoting the direction of the convection and Ω in is the inflow boundar of Ω such that the boundar of Ω is Ω = Ω in Ω out. The inflow and outflow boundaries are defined b b n < on Ω in and b n > on Ω out, respectivel. Considering the boundar of the whole domain Ω, we split also this boundar in an upwind and downwind boundar such that Ω = Ω in Ω out. Then, according to the Lagrange multiplier theorem

18 4. Weak forms for embedded boundar conditions 6 Error on domain of interest e e e 8.5e Figure : The approimate solution u = + of u =, and the error on the domain Ω Ω 2 for a fourth order combined hbrid-smmetric DG discretisation with embedded circle segment Dirichlet boundar condition (R =3/4). we arrive for the boundar value problem at the following weak formulation: find u H ( Ω) and χ H /2 ( Ω in ) such that v bu d+ n b χv ds + n b uv ds + n b qu ds (4.7) Ω Ω in Ω out Ω in = fv d+ n b qu ds, v H ( Ω), q H /2 ( Ω in ), Ω Ω in in which we assume that u on the fictitious domain Ω satisfies the differential equation and is the continuation of the solution u on the domain Ω. Figure 2 shows the solution and the error if (4.7) is used as the starting point for a discretisation with cubic polnomials, as discussed above. Error on domain of interest 8e 8 6e 8 4e 8.5 2e 8 2e e Figure 2: The approimate solution u = of the convection equation b u =andthe error on the domain Ω for a fourth order hbrid discretisation with embedded circle segment Dirichlet boundar condition (R =8/).

19 4. Weak forms for embedded boundar conditions Two adjacent cells with a common interior embedded boundar condition In this section we stud a finite element discretisation of the convection diffusion equation ε u + b u =, (4.8) discretized on two adjacent cells Ω and Ω 2, with vertices (, ), (, ), (, ), (, ) and (, ), (, ), (, ), (, ) respectivel. The embedded Dirichlet boundar condition is given on the half circle = R 2,, so that the domain of interest is given b Ω=( Ω Ω 2 ) \ { (, ) R 2}. We first consider the diffusion part of the equation. Then, the weak hbrid formulation of the problem reads: find u H ( Ω h )andp H /2 ( Ω Γ int ) such that: ( u, v) Ωh p, v Ω p, n [v] Γint q, u Ω q, n [u] Γint (4.9) =(f,v) Ωh q, u Ω, v H ( Ω h ), q H /2 ( Ω Γ int ), where H ( Ω h ) is the broken Sobolev space on Ω Ω 2 and the common interface is given b Γ int = Ω Ω 2,andnis the normal vector. The interface Γ int, not including the fictitious part, is defined b Γ int = Ω Ω 2. Recognizing in a normal flu on the common p Γint interface Γ int, we define the trace operators, γ Ω : H ( Ω) C ( Ω) H /2 ( Ω Γ int )and γ Ω : H ( Ω) C ( Ω) H /2 ( Ω Γ int ). In order to approimate the normal derivatives on Ω Γ int we proceed as in Section 4. and introduce the polnomial subspace S h ( Ω) H ( Ω) C ( Ω). We split this space as: with S h ( Ω) = Q h ( Ω) K h ( Ω), K h ( Ω) = ker( γ Ω ) S h ( Ω). Then the discrete version of (4.9) reads: find u h S h ( Ω h )andχ h Q h ( Ω) such that ( u h, v h ) Ω h γ Ω (χ h ),γ (v h ) γ Ω Ω (χ h ), [v h ] (4.2) Γ int γ Ω (q h ),γ (u h ) Ω γ Ω (q h ), [u h ] Γ int =(f,v h γ Ω ) Ωh (q h ),u v Ω h S h ( Ω h ),q h Q h ( Ω). (Notice the polnomial spaces used!) For the polnomial space S h ( Ω h ) we can take the usual space of piecewise cubic polnomials in each coordinate direction on the partitioning Ω Ω 2. On the other hand, it is not trivial to find a cubic polnomial space for Q h ( Ω) H ( Ω) C ( Ω). As we do not want to make our discretisation unnecessar complicated and epensive, we eliminate the etra degrees of freedom for χ h b identifing them with u h. Similarl, identifing q h and σ v h on Ω Γ int we arrive at a discontinuous Galerkin discretisation.

20 5. A singularl perturbed PDE on onl two cells with a half circle ecluded 8 As our first interest is an efficient fourth order discretisation, we epect it to be highl improbable that instabilit will occur in the discrete operator, because the one-dimensional eperiment shows onl a single pole. Nevertheless, the use of DG discretisation forces us to monitor for possible singularities. Now the DG version of (4.2) is: find u h S h ( Ω h ) such that ( u h, v h ) Ωh u h,v h Ω u h, [v h ] Γint + σ v h,u h Ω + (4.2) σ v h, [u] Γint =(f,v h ) Ωh + σ v h,u Ω, v S h ( Ω h ), with the usual choices for the normal flu functions. Net we consider the convection part of (4.8). equation So, on the domain Ω we consider the b u = f, u = u on Ω in. (4.22) For simplicit we set b =(, ). Then the embedded boundar is an outflow boundar for Ω, whereas for Ω 2 it is an inflow boundar. Hence, we can neglect this embedded boundar in Ω, whereas in cell Ω 2 we must introduce a Lagrange multiplier in order to satisf the upwind boundar condition on the circle bow. Hence, we arrive at the following weak form for the convection part: find u, χ H ( Ω h )=H ( Ω Ω 2 ) v bud+ (n bu)vds + (n bχ)vds + (n bu)vds Ω Ω,out Ω 2,in Ω 2,out + (n bq)uds (n bq)u ds (4.23) Γ int Γ int = fvd (n bu )vds + (n bq)u ds Ω Ω,in Γ D v, q H ( Ω), where u = u Ω = u Γint. If we want to eliminate in (4.23) the etra degrees of freedom, we set χ = u and q = v. Linear combination of (4.2) and (4.23) gives a discretisation of the convection diffusion equation u + b u = f in Ω, u = u on Ω. (4.24) The Figures 3 and 4 show the solution and the error of a discretisation of (4.24) b means of (4.2) and (4.23) with tensor-product cubics as approimation and test spaces. 5. A singularl perturbed PDE on onl two cells with a half circle ecluded In this section we are interested to solve the following convection diffusion problem. (see Figure 5) More details about this problem can be found in [5] ε u + u = f on Ω ={ (, ) <<, <<}, (5.) u = on Ω ={ (, ) =, <<; <<, =}, u = onγ D = { (, ) = R 2, >; R< }, (5.2) n ε u = onγ N = { (, ) R< < } {(,) <<}.

21 5. A singularl perturbed PDE on onl two cells with a half circle ecluded 9 Error on domain of interest 3.5e 8 2 e 8 5e 9 5e 9 e 8.5e Figure 3: The approimate solution u = of u + u = f andtheerroronthe domain Ω = Ω Ω 2 for a fourth order smmetric-dg discretisation with embedded circle Dirichlet boundar condition (R =3/4). Let Ω and Ω 2 be two unit cells with respectivel vertices (, ), (, ), (, ), (, ) and (, ), (, ), (, ), (, ) so that Ω = Ω Ω 2. For this problem we want to stud the smmetric and the Baumann-Oden DG-method. First we stud the diffusion part of (5.) and replace the homogeneous Neumann boundar condition on Γ N = { (, ) =, <<} with the homogeneous Dirichlet boundar condition, in order to obtain a problem smmetric around =. Now the corresponding hbrid formulation (2.7) for (5.) reads: find u H ( Ω h )andp H /2 ( Ω Γ int ) such that ( u, v) Ω p, v Ω q, u Ω Γ D p, n [v] Γ int q, n [u] Γint = (5.3) =(f,v) q, u Ω ΓD, v H ( Ω h ),q H /2 ( Ω Γ int ). Here H ( Ω h ) is the broken Sobolev space on Ω Ω 2 and the jump operator is given b [v] = n v + n Ω 2 v. Further, Γ Ω2 int = Ω Ω 2 is the interior wall on which the true solution is continuous. However, continuit is not required outside Ω and hence not on all of Γ int = Γ int Ω. To arrive at the DG-discretisation of (5.3) we take for the test and trial space, S h ( Ω) H ( Ω h ), the tensor product of polnomials of degree p<4ineach of the coordinate directions and we write for the approimation u h = c e,i,j φ e,i () φ e,j (). (5.4) e<2 i,j<4 In practice we construct a basis from (4.5). Net, for the DG discretisation, we eliminate the etra equations and degrees of freedom for the Lagrange multiplier using the fact that p represents the normal flu of u at Ω andat the internal wall Γ int. So replacing p b n u h on Ω andb n u h on Γ int and replacing

22 5. A singularl perturbed PDE on onl two cells with a half circle ecluded 2 Error on domain of interest Figure 4: The approimate solution u = e + of u + u = f andtheerroronthe domain Ω Ω 2 for a fourth order smmetric-dg discretisation with embedded circle Dirichlet boundar condition (R =3/4). Γ D Γ D Ω Γ D Γ int Ω2 Γ N Γ N Γ N Figure 5: The domain for problem (5.-5.2). similarl q b σn v, the DG discretisation of (5.3) reads: find u h S h ( Ω) such that ( u h, v h ) Ω n u h,v h Ω + σ n v h,u h Ω ΓD u h, [v] Γ int (5.5) +σ v h, [u] Γint =(f,v h )+σ n v h,u Ω ΓD, v h S h ( Ω). Figure 6 shows the solution of the smmetric (σ = ) and Baumann-Oden (σ = ) discretisation. We see that both solutions are smmetric indeed, because of the smmetric structure of the problem. On the other hand we recognize the instable behavior of the smmetric DG method, which is of poor qualit compared with the solution of the Baumann-Oden method. We continue b considering both methods for the diffusion part of the equation and boundar conditions as in (5.2). Then the discrete formulation is also given b (5.5), ecept that the Dirichlet boundar condition at {(,) <<} is replaced b the homogeneous Neumann boundar condition n u =. The corresponding solutions of the smmetric and Baumann-Oden method are shown in Figure 7. Now the solutions are not smmetric. Again, the solution of the smmetric DG-method is poor compared to the solution of the Baumann-Oden DG method. Finall, we take the convection part of (5.). The Lagrange weak formulation of the

23 5. A singularl perturbed PDE on onl two cells with a half circle ecluded 2 convection term for the cell Ω 2 reads: find u H ( Ω 2 )andp H /2 ( Ω in ) such that ( v + b,u) Ω2 vn bu ds+ p n b,v Ω Ωin =, v H ( Ω 2 ), 2 (5.6) qn b,u = qn b,u, q H /2 ( Ω in ), with b =(, ) and u = on the embedded circle, Γ D Ω in, while u = u on the common interior boundar, Γ int Ω in,withu the upwind value of u obtained from Ω. As for the diffusion term, we can rewrite (5.6) in a hbrid formulation, where the Lagrange multiplier is computed on Ω in, the inflow edge of the domain Ω 2. Thus, we obtain: find u H ( Ω 2 )and p H /2 ( Ω in ) such that ( v b,u) Ω 2 + vn bu ds+ p n b,v Ω in =, v H ( Ω 2 ), (5.7) Ω 2 qn b,u Ωin = qn b,u Ωin, q H /2 ( Ω in ). Net, the second term in (5.7) is split in an integration part over the inflow and a part over the outflow edge of Ω 2 so that we can combine the integration of u and p over the inflow wall Ω in. This ields: find u H ( Ω 2 )andp H /2 ( Ω in ) such that ( v b,u) Ω2 + ( p + u) n b,v Ωin + u n b,v Ωout + qn b,u Ωin (5.8) = qn b,u Ωin, v H ( Ω 2 ), q H /2 ( Ω in ). Writing p = p + u, this is simplified to: find u H ( Ω 2 )andp H /2 ( Ω in ) such that ( v b,u) Ω 2 + p n b,v Ω in + u n b,v Ω out + qn b,u Ωin (5.9) = qn b,u Ωin, v H ( Ω 2 ), q H /2 ( Ω in ). To eliminate the Lagrange multiplier p, we recognize this function as the value of u at the boundar Ω in. The corresponding equations are eliminated b taking q = σv on Ω in, ielding the DG-formulation of the convection term: find u H ( Ω 2 ) such that ( v + vn bu ds+ σ v n b, ( u u ) (5.) b,u) Ω2 Ω Γ int 2 +σ v n b,u ΓD = σ v n b,u ΓD, v H ( Ω 2 ). In cell Ω the embedded boundar is an outflow boundar and, hence, for the convection part gives no boundar condition. Therefore, we ma treat Ω as a normal convection DG-cell. The discretisation of problem (5.) is obtained b combining (5.5), (5.8), and (5.): find u h S h ( Ω) such that (ε u h, v h n ε u ) Ω h,v h + σ n ε v Ω h,u h Ω ΓD ε u h, [v h (5.) ] Γint +σ ε v h, [u h ] Γint ( v h b,u h + v ) Ω h n b,u h + v Ω,out h n bu h ds Ω 2 +σ v h n b, ( u h u ) h + σ v Γ int h n b,u h ΓD Ω 2 =σ n ε v h, ΓD + σ v h n b, ΓD Ω 2, v h S h ( Ω).

24 6. Conclusion 22 Figure 8 shows the solutions of the fourth order discretisation of (5.) for R =3/, for the different values of ε =,.,.2,.. We see that in all cases the solution is stable. For values of ε = O() we see clearl the approimation of the boundar condition u = on the circle bow, while for small values of ε, when the true solution shows a thin boundar laer, tpical effects of the weak boundar requirement show up. Figure 9 shows the solution for ε =/5. Although it seems that the solution is not capable to catch the boundar laer in Ω at the upwind side of the circle, we clearl see a boundar laer arise in the fictitious part of the domain if we consider the total Ω. Clearl, the cubics are not able to represent the thin circular boundar laer. Notice in particular, at =,<R, the discontinuit in the fictitious part Ω. For small values of ε<< the boundar laer disappears smmetric DG Baumann-Oden DG Figure 6: The approimate solution u h of u = on the domain Ω with smmetric boundar conditions and fourth order discretisations with embedded circle Dirichlet boundar condition (R =3/). 6. Conclusion In this paper we propose a technique for the treatment of second-order elliptic PDEs with comple Dirichlet boundar conditions in combination with discontinuous Galerkin discretisation. The aim is to maintain a structured, regular rectangular grid while solving problems with irregular curved boundar conditions. The comple domain on which the solution is sought, is covered b a fictitious domain with the structured, regular rectangular grid. An embedded boundar is the transition between the domain of interest and the fictitious part of the computational domain. We present and compare several weak forms for the diffusion part of the equation: the Lagrange multiplier form, the hbrid form and the DG-form. The hbrid form shows stabilit for an arbitrar location of the embedded boundar, whereas for the other forms the discretisation ma become instable for particular locations of the Dirichlet BC. The problem is studied, first for a single cell and then for a couple of adjacent cells, either sharing or not sharing the embedded boundar. We also describe the treatment of a convection part in the equation. As an eample, we solve a singularl perturbed boundar value problem on a comple do-

25 6. Conclusion smmetric DG Baumann-Oden DG Figure 7: The approimate solution u h of u = on the domain Ω with the boundar conditions (5.2) for a fourth-order smmetric and Baumann-Oden discontinuous Galerkin discretisation. The embedded circle Dirichlet boundar condition is located at R =3/. main b means of a fourth-order DG-discretisation on onl two cells. Although as epected it appears to be impossible to accuratel represent sharp boundar laers with a comple structure b means of a few cubic polnomials, the boundar condition treatment is quite effective in handling such comple situations.

26 24 6. Conclusion ε=.5.2 ε = / ε = / ε = / Figure 8: The approimate solution uh of ε u + u =, on the domain eterior of the circle for a fourth order Baumann-Oden DG discretisation with Dirichlet boundar condition u = on the circle, (R = 3/). References. D.N. Arnold, F. Brezzi, B. Cockburn, and D. Marini. Unified analsis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer. Anal., pages , R. Cortez and M. Minion. The Blob projection method for immersed boundar problems. Journal of Computational Phsics, 6: , D. Goldstein, R. Handler, and L. Sirovich. Modeling a no-slip flow boundar with an eternal flow field. J. Comp. Phs., 5: , W. Hackbusch. Theorie und Numerik elliptischer Differentialgleichungen. Teubner Studienbuecher, P. W. Hemker. A singularl perturbed model problem for numerical computation. Journal of Comp. and Appl. Math., 76: , P.W. Hemker, W. Hoffmann, and M.H. van Raalte. Two-level fourier analsis of a multigrid approach for discontinuous galerkin discretisation. Technical Report MAS-R26,

27 25 6. Conclusion Solution on total domain Total domain.8.5 Domain of interest Figure 9: The approimate solution uh for the same problem as in Figure 8, with ε = /5. At the right, the approimate solution on Ω, the domain of interest. Left, the approimate the whole domain where the approimation is constructed, including the solution on Ω, fictitious part. CWI, Amsterdam, April 22. submitted to SIAM SISC. 7. D. W. Hewett. The embedded curved boundar method for orthogonal simulation meshes. J. Comp. Phs., 38:585 66, D.W. Hewett and C.S. Kuen. The dielectric boundar condition for the embedded curved boundar (ECB) method. Technical Report UCRL-JC-2973, Lawrence Livermore Nat. Lab., 998. presented at the 6th International Conference on the Numerical Simulation of Plasmas, Februar 998, Santa Barbara, CA. 9. Jungwoo Kim, Dongjoo Kim, and Haecheon Choi. An immersed-boundar finite-volume method for simulations of flow in comple geometries. Journal of Computational Phsics, 7:32 5, 2.. C.S. Kuen. Embedded curved boundaries and adaptive mesh refinement. Technical Report UCRL-JC-29729, Lawrence Livermore Nat. Lab., R.J. LeVeque and Zhilin Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM Journal on Numerical Analsis, 3(4):9 44, R.J. LeVeque and Zhilin Li. Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM Journal on Scientific Computing, 8(3):79 735, Z. Li. The Immersed Interface Method - A Numerical Approach for Partial Differential Equations with Interfaces. PhD thesis, Universit of Washington, J. Nitsche. U ber ein Variationsprinzip zur Lo sung von Dirichlet Problemen bei Verwendung von Teilra umen die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg, 36:9, 97.

28 6. Conclusion C.S. Peskin. Numerical analsis of blood flow in the heart. J. Comp. Phs., 25:22 252, C. Tu and C. S. Peskin. Stabilit and instabilit in the computation of flows with moving immersed boundaries: A comparison of three methods. SIAM Journal on Scientific and Statistical Computing, 3:36 376, 992.