Comput. Methods Appl. Mech. Engrg.

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1 Coput. Methods Appl. Mech. Engrg. 200 (2011) Contents lists available at ScienceDirect Coput. Methods Appl. Mech. Engrg. journal hoepage: A posteriori error analysis for a cut cell finite volue ethod Don Estep a,b, Michael Pernice c, Sion Tavener a, Haiying Wang d, a Departent of Matheatics, Colorado State University, Fort Collins, CO 80523, USA b Departent of Statistics, Colorado State University, Fort Collins, CO 80523, USA c Idaho National Laboratory, Idaho Falls, ID 83415, USA d Departent of Matheatical Sciences, Michigan Technological University, Houghton, MI 49931, USA article info abstract Article history: Received 23 Deceber 2009 Received in revised for 30 July 2010 Accepted 22 Noveber 2010 Available online 26 Noveber 2010 Keywords: Cut cell proble Discontinuous diffusion A posteriori error analysis Adjoint proble Finite volue ethod Modeling error We study the solution of a diffusive process in a doain where the diffusion coefficient changes discontinuously across a curved interface. We consider discretizations that use regularly-shaped eshes, so that the interface cuts through the cells (eleents or volues) without respecting the regular geoetry of the esh. Consequently, the discontinuity in the diffusion coefficients has a strong ipact on the accuracy and convergence of the nuerical ethod. This otivates the derivation of coputational error estiates that yield accurate estiates for specified quantities of interest. For this purpose, we adapt the well-known adjoint based a posteriori error analysis technique used for finite eleent ethods. In order to eploy this ethod, we describe a systeatic approach to discretizing a cut-cell proble that handles coplex geoetry in the interface in a natural fashion yet reduces to the wellknown Ghost Fluid Method in siple cases. We test the accuracy of the estiates in a series of exaples. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction 1.1. Elliptic probles with discontinuous diffusion coefficients and cut-cell discretizations In this paper, we consider the solution of a diffusive process: r ðarpþ ¼f ; in X; p ¼ g; on ox; in a convex polyhedral doain X 2 R 2 with boundary ox that consists of two distinct aterials with different aterial properties, see Fig We assue that the diffusion value changes discontinuously across a sooth interface curve C d interior to X, while we require that the solution be continuous and have a continuous noral flux across the interior interface boundary. Specifically, we assue that a is sooth in each sub-doain of X deterined by C d, a has one-sided liits at C d, and a(x) is bounded below by a positive nuber. We also assue that f 2 L 2 (X) and g 2 H 1/2 (ox). Finally, we assue p 2 W = L 2 (X) and u = arp 2 V = H (div;x)={v 2 (L 2 (X)) 2 : divv 2 L 2 (X)}. Such probles arise in any contexts, e.g. Corresponding author. Tel.: ; fax: E-ail address: haiyingw@tu.edu (H. Wang). ð1þ biology, cheistry, aterial science, and fluid flow. The particular proble otivating this work is heat conduction in nuclear fuel rods consisting of an uraniu core and a steel cladding. Elliptic probles with discontinuous diffusion coefficients pose well known challenges for reliably accurate nuerical solution because of the difficulties arising fro the discontinuity. Consequently, this is a very active area of nuerical analysis in which a nuber of approaches are being pursued, e.g. special discretization ethods, adaptive esh refineent, ebedded interface ethods, and so forth. The difficulties are agnified significantly when the interface has coplex geoetry that is not aligned with a discretization esh and when the coefficient is not piecewise constant. Generally in this case, loss of order of accuracy is expected and nuerical error in coputed results is always significant. Consequently, obtaining accurate estiates of the error in a quantity of interest coputed fro a given nuerical solution is critically iportant in practical applications regardless of the particular nuerical approach. This provides the basic otivation for the work in this paper. In any application doains involving diffusive and transport probles, there is a strong preference towards the use of locally conservative ethods, e.g. finite volue as well as special discontinuous Galerkin and ixed finite eleent ethods, even at the cost of relatively low order of accuracy for soe coon ethods. Since locally conservative schees are generally easier to construct and ipleent on regularly-shaped discretizations, /$ - see front atter Ó 2010 Elsevier B.V. All rights reserved. doi: /j.ca

2 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) Fig Two adjoining aterials with different diffusion properties. The sooth interface cuts through the cells of a regular discretization. there are several well known approaches for probles with discontinuous coefficients that eploy regularly-shaped discretizations, so that the interface cuts through the discretization cells, yielding a cut-cell proble. In this paper, we study the well known Ghost Fluid Method for discontinuous interface probles [1,2]. The Ghost Fluid Method is a special finite volue ethod introduced for the related proble in which flux and state ay be discontinuous across the interface. The Ghost Fluid Method yields a locally conservative finite volue schee that approxiates discontinuous solutions without oscillations near the interface. It has been successfully used in a wide range of applications, including spray atoization [3,4], droplet dynaics [5 7], cobustion [8,9], vaporization at reactive interfaces [10,11], bubbly flows [12], aluina electrolysis [13], and visualization of fluid flow [14,15]. These applications all ipleent the level set ethod [16] to describe a sharp interface between fluid states and to track coplex topological changes of the interface. Alternatively, one could eploy discretizations that are atched to the interface. However, such ethods also face considerable technical difficulties in analysis and ipleentation. In particular, generating high quality unstructured and body-fitted ulti-block grids for coplex interface geoetries is not yet fully autoated, and can consue a large fraction of the tie to perfor a siulation. On the other hand, generating a cut-cell representation of an interface is uch faster and generally ore robust, since it relies on solving local intersection probles, a procedure that is fully autoated [17]. Moreover, there are situations that suggest the use of regular discretizations, e.g. Evolution probles in which the interface is evolving, perhaps dependent on the behavior of the solution, in which case repeatedly generating eshes to atch the evolving interface is expensive. Probles in which the interface is deterined fro experiental easureent and is described with relatively poor accuracy, in which case using a high resolution representation is not justified. Sensitivity analysis studies involving the solution of any interface probles with varying interfaces, in which case it ay be desirable to iniize the cost of each saple solution. Probles in which the quantities of interest to be coputed fro the odel are relatively insensitive to the exact location of the interface The connection between finite volue and finite eleent ethods and accurately estiating the error in a cut-cell discretization Producing accurate estiates of nuerical error in coputed results for a discontinuous interface proble is a practically iportant issue, e.g. for uncertainty quantification. A priori convergence analysis for a cut-cell discretization is probleatic and the results, typically involving loss of order, are discouraging. Indeed, for the general situation, the convergence analysis does not yield an order of convergence, see [2]. Instead, we take a different approach to error estiation by deriving a coputational error estiate that provides a way to copute an accurate estiate of the error in a quantity of interest during an actual coputation. For this purpose, we adapt a powerful a posteriori error analysis technique well-known in the finite eleent counity. This approach [18 23] uses an adjoint proble, coputable residuals, and variational analysis to produce a coputational estiate of the error in a quantity of interest deterined by a specified linear functional of the solution. The resulting estiate is generally accurate and is also precise in the sense of decoposing the error into contributions fro various sources, taking into account the full effects of stability on accuulation, cancellation, and propagation of errors. This a posteriori approach is based on variational analysis, which akes it natural for a finite eleent discretization. To apply the ethod to a finite difference or finite volue discretization, these discretizations are described as a particular kind of finite eleent ethod eploying specialized quadrature forulas inside cells to evaluate integrals in the weak for of the differential equation. For exaple, there is a well-known equivalence between ixed finite eleent ethods eploying special basis functions and quadrature forulas and cell-centered finite volue ethods [18,24 26]. We use this in [18] to derive a posteriori error estiates for cell-centered finite volue ethods for elliptic probles. It is straightforward to use this approach when the interface aligns with the cell boundaries in the discretization esh. In particular, there are no theoretical issues in using quadrature to evaluate integrals involving the diffusion coefficient inside esh cells. However, a cut-cell proble is probleatic because the use of quadrature forula across a cell in which the diffusion paraeter is discontinuous is not justified theoretically. Likewise, actually coputing the integrals exactly when there is a coplex interface geoetry inside a cell is very probleatic. See [27] for a recent overview of approaches for treating this lack of soothness. Thus, a second ajor goal of this paper is deriving a systeatic approach to finite eleent discretization that applies to general cut-cell discretizations, reduces to the Ghost Fluid Method in siple cases, and allows the application of a posteriori error analysis. We stress that our interest lies in the general proble of an interface for non-constant coefficient diffusion with coplex geoetry that does not align with the discretization. It is generally probleatic to extend standard approaches for treating discontinuous interface probles based on siplifying assuptions, such as interfaces that align with cell boundaries. Our approach to a cut-cell finite eleent ethod involves two steps. First, we replace, or odel, the discontinuous diffusion coefficient with a piecewise polynoial function (linear in one diension and biquadratic in two diensions) in a relatively sall region X d containing the interface C d that scales with the esh. The approxiation is continuous in X d. This essentially spreads the discontinuity to the boundary of X d, so that the lack of soothness of the new odel coefficient is restricted to cell boundaries. After this, we choose an appropriate basis and quadrature forula to obtain a finite volue schee that is equivalent to the Ghost Fluid Method in siple cases. To describe this in a siple case, consider a one diensional proble with a piecewise constant diffusion where the discontinuity is located at a point x in the interior of a cell, see Fig The Ghost Cell approxiation is deterined by a standard finite volue ethod applied to a new odel proble in which the value

3 2770 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) Fig Left: We illustrate the coputation of the weighted haronic average in the case of a piecewise constant diffusion function. The discontinuity is at x. The weighted haronic average value a iþ1=2 at x i+1/2 is deterined by the values at the cell centers x i, x i+1 containing x weighted by the distances fro x to the cell centers. We also plot the piecewise linear continuous odel diffusion a (x) in one diension. a and a agree outside the interval X d =[x i1/2,x i+3/2 ]. Right: The continuous, piecewise biquadratic odel diffusion coefficient in two diensions for a typical proble. The vertical lines indicate the nodal points for the biquadratic interpolation. The soothness within cells and the continuity across the cell boundary is evident. of the diffusion at the cell boundary point closest to the discontinuity x is altered to be a weighted version of the haronic average 1 2 ð1 þ 1Þ 1 of the values a and b at two points. We discuss the a b weighted haronic average (3) and its iportance for discontinuous interface probles below. We note that there generally is no unique way to write a given finite volue ethod as an equivalent finite eleent ethod. In the course of deriving the ethod presented in this paper, we derived a nuber of approaches for the Ghost Fluid Method. However, the approach described in this paper deals naturally with soe geoetric issues that arise in two diensions and allows for coputation of accurate a posteriori error estiates. As entioned, our approach involves searing or soothing the discontinuity on the interface so that any lack of soothness in the odel diffusion lies on the cell boundaries of the sall region X d. This region shrinks with the esh size, and the odel coefficient converges to the true coefficient as the esh is refined. A key point is that the Ghost Fluid Method involves both a odeling step-in the for of altering the value of the diffusion in a sall region-and a discretization step-in the for of ixed finite eleent ethod with special quadrature. Both steps contribute significantly to the error of the ethod. In this application, we show how to odify the standard analysis for a ixed finite eleent/finite volue ethod [18] to account for the odeling error in the cut-cell finite eleent ethod. This provides the tools to estiate the relative contributions to the error fro odeling and discretization. The a posteriori error analysis used in this paper can be carried out generally for a wide range of discretizations. The critical step is to identify the two key ingredients: (1) odeling the diffusion and (2) introducing an appropriate ixed finite eleent or discontinuous Galerkin ethod Outline of the paper The outline of the paper is as follows. In Section 2, we describe the discretization. We present the results of an analysis of the odeling error along with a goal-oriented a posteriori error analysis in Section 3 while the proofs are presented in Appendix A. In Section 4, we present a series of nuerical exaples. In Section 5, we present an analysis of the effects of error in the location of the interface. In Section 6, we present the conclusion. 2. Construction of the discrete approxiation Recall that the goal is to construct a finite eleent discretization that is equivalent to the Ghost Fluid Method in siple cases, yet allows treatent of interfaces with coplex geoetry and non-constant diffusion coefficients. We discretize (1) using a two step process. We first replace (1) by a odel proble. Then, we discretize the odel proble using a ixed finite eleent ethod Construction of the odel proble The first step in the discretization is to replace (1) by a odel proble: r ða rpþ ¼f ; in X p ¼ g; on ox; ð2þ where we replace a a. To define a, we use a weighted haronic average coputed along lines. Consider a canonical interval X on a line. The line contains the proble doain in one diension and joins neighboring cell centers in two diensions. We assue that the interval has been discretized using a partition {[x j1/2,x j+1/2 ], j =1,2,...,N}, with cell boundaries located at {x j+1/2,j =0,1,...,N}, cell centers {x j,j =1,2,...,N}, and X =[x 1/2,x N+1/2 ]. We suppose the discontinuity is located at x 2 [x i,x i+1 ], see Fig We define the weighted haronic average a iþ1=2 of the values of a at x i+1/2 to be: a iþ1=2 ¼ c þ 1 c 1 ; ð3þ a i a iþ1 where a i = a(x i ),a i+1 = a(x i+1 ) and c =(x x i )/(x i+1 x i ). That is, a iþ1=2 is a haronic average weighted by the fractions of the sub-doain over which a takes its different values Construction of a in one diension Our focus is on the two diensional case, but the one-diensional case is illustrative. We define a odel proble in which discontinuous diffusion coefficient a(x) is replaced by a continuous piecewise linear approxiation a (x) that interpolates a at x i1/2 and x i+3/2 and has value equal to the weighted haronic average at x i+1/2. This eans that we alter the value of a to obtain a in the region X d =[x i1/2,x i+3/2 ], and set a a in the copleent. On X d,a (x) is given by

4 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) Fig Illustration of Algorith 1. (a) Left: Beginning with an interface and a discretization. Middle: We identify the odeling region and set values at the cell centers for cells in the region. Right: We then copute haronic averages at the appropriate cell boundary centers. (b) Left: We copute standard averages at the other interior cell boundary centers. Middle: We use appropriate averages of haronic and standard averages deterined along the diagonal collinear connecting line segents between cell centers to deterine values at interior cell nodes. Right: We use interpolation to assign the reaining values on the boundary of the odeling region. ðxþ ¼ 8 1 a a iþ1=2 ð i1=2 Þ 1 x iþ1=2 x i1=2 x x i1=2 >< x 2 x i1=2 ; x iþ1=2 ; 1 ða iþ3=2 Þ 1 a iþ1=2 x x x iþ3=2 x iþ3=2 iþ1=2 >: x 2 x iþ1=2 ; x iþ3=2 ; 1 þ ai1=2 ; 1 þ aiþ3=2 ; where a iþ1=2 is the weighted haronic average (3). We illustrate ðxþ in Fig Construction of a in two diensions It is probleatic to extend the one diensional odel approxiation directly to two diensions because of increased geoetric coplexity. In one diension, cell centers are connected collinearly through a cell node, while in two diensions, cell centers are connected collinearly through a cell boundary at the cell boundary center. Consequently, we treat cell boundary centers and cell nodes differently in two diensions. To create the odel coefficient, we again use the weighted haronic average (3), but now applied on various line segents that connect cell centers in neighboring cells that share a coon cell boundary or a coon cell node diagonally. We discretize the canonical two-diensional doain X which we take to be a unit square. We partition X in the x and y-directions as 0 ¼ x 1=2 < x 1 < x 3=2 < x 2 < < x k1=2 < x k < x kþ1=2 ¼ 1; 0 ¼ y 1=2 < y 1 < y 3=2 < y 2 < < y 1=2 < y < y þ1=2 ¼ 1: We then define the cells (finite volues) to be the rectangles: K ij ¼ x i1=2 ; x iþ1=2 yj1=2 ; y jþ1=2 ; i ¼ 1;...; k; j ¼ 1;...; with the centers (x i,y j ) and nodes of half indices. We set: Dx iþ1=2 ¼ x iþ1 x i ; i ¼ 1;...; k 1; Dx i ¼ x iþ1=2 x i1=2 ; i ¼ 1;...; k; Dy jþ1=2 ¼ y jþ1 y j ; j ¼ 1;...; 1; Dy j ¼ y jþ1=2 y j1=2 ; j ¼ 1;...; : The discrete esh X h is then defined as X h ={K ij,i =1,...,k; j =1,..., }. To deterine a odel diffusion coefficient, we construct a region X d using discretization volues that contain the interface curve C d in the interior together with discretization volues that share a cell boundary with one of these cut cells. We alter the value of a to obtain a in the region X d and set a a in the copleent. We choose the odel coefficient a to be a continuous piecewise biquadratic function on X d, with interpolation points at the four cell nodes (vertices), the four cell boundary centers and at the cell center. This construction is illustrated for a pair of adjacent cells in X d in Fig The values of the coefficients for the odel proble at nodes and cell faces are deterined according to Algorith 1 and a piecewise biquadratic surface is then fit to this data. Clearly nuerical cell-by-cell quadrature forulae are now well behaved for the odel proble. We describe the procedure for constructing a in Algorith 1. The Algorith is illustrated in Fig. 2.2.

5 2772 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) Algorith 1: Construction of the cut-cell odel diffusion coefficient 1. Deterine X d. Identify the cut cells containing the interface C d. Add those cells that share a cell boundary with the cut cells to obtain X d. Deterine the location and orientation of the interface with respect to each cut cell center 2. Set values at cell centers in X d. In case the diffusion is constant in a cell, use that constant value. If the diffusion varies over a cell, use the integral average over the cell 3. Set values at cell boundary idpoints in X d. For a cell boundary idpoint that lies on a line segent connecting a pair of cell centers in X d, use the weighted haronic average (3) of the cell center values if the line segent connecting the cell centers is cut by the interface, otherwise use the regular average of the cell center values 4. Set values at cell nodes in X d. We copute two values for each cell node and then we average to obtain the value for the cell node. Each value is obtained by averaging cell center values for each pair of cell centers that lie on a diagonal line segent through the cell node. We use the weighted haronic average (3) when the interface cuts through the diagonal line segent joining the two cell centers. Otherwise, we use the regular average 5. Set the values at the reaining interpolation points in X d. Use interpolation to set any reaining values in the interpolation points for the piecewise biquadratic odel diffusion, e.g. at points located on the boundary of X d Note that in two diensions, we replace the interface of the discontinuity in a by a stepwise interface ox d which is coprised of cell boundaries while the diffusion is replaced in a region bounded by the new interface. Since X d scales with the esh size, the odel diffusion converges to the true diffusion as the esh is refined Discretization of the odel proble With the introduction of a odel diffusion that re-locates any discontinuity to cell boundaries, we can now discretize the odel proble using a ixed finite eleent ethod ipleented with a specially chosen quadrature that yields a cell-centered finite volue schee [18,24 26]. We set u = a rp and assuing that: p 2 W ¼ L 2 ðxþ; u 2 V ¼ Hðdiv; XÞ; we replace (2) by the equivalent first order variational syste: u ;v X ð p ; r vþ X ¼hg;vni ox ; ð4þ ðr u ; wþ X ¼ ðf ; wþ X ; for any (v,w) 2 (V,W), where (,) D and h,i c denote inner products on D R 2 and the boundary c. We discretize (4) using a Raviart Thoas ixed finite eleent schee [28,29]. We choose the finite eleent spaces W h W, consisting of the space of piecewise constant functions, and V h V, which is the space of vector-valued functions whose x-coponents are continuous linear in x and discontinuous constant in y and whose y-coponents are discontinuous constant in x and continuous linear in y. In addition, we apply quadrature to evaluate the integrals in (4). We use subscripts T x and T y to denote the trapezoidal quadrature rules in the x and y-direction respectively, subscripts M x and M y to denote the idpoint quadrature rules in the x and y-direction respectively, and subscript M to indicate the idpoint approxiation on cell edges. Further details about these choices can be found in [18]. The approxiation becoes: copute (u,h,p,h ) 2 V h W h satisfying: ux ;h ;vx þ uy ; ;h vy p ;h ; r v ¼hg;vniM ; T xm y M xt y ðr u ;h ; wþ ¼ðf ; wþ MxMy ; for any v 2 V h,w 2 W h, where we set u ;h ¼ðu x ;h ; uy ;h Þ;v ¼ðvx ;v y Þ. The finite eleent discretization for a one-diensional proble is the natural restriction of the two diensional ethod. We do not give the discrete forulas here Equivalence to the (Ghost Fluid) cell-centered finite volue schee If we consider the discrete equations defining the finite eleent pressure approxiation, using p to denote p,h, we obtain: p iþ1;j p i;j p i;j p i1;j Dy j a ;iþ1=2;j a ;i1=2;j Dx iþ1=2 Dx i1=2! Dx i a ;i;jþ1=2 p i;jþ1 p i;j Dy jþ1=2 a ;i;j1=2 p i;j p i;j1 Dy j1=2 ¼ f ij Dx i Dy j ; for 1 6 i 6 k, 1 6 j 6, using proper boundary values. These are the discrete equations for the cell-centered finite volue schee Observations about the hybrid discretization 1. Both the one-and two-diensional odel diffusions are convex interpolants, and hence they preserve any positivity condition placed on the original diffusion constant. 2. The approxiation conserves flux across cell boundaries at the cell boundary centers (which is the sense in which standard finite volue ethods are conservative) and at cell nodes across diagonals in an average sense if the node does not lie on C d. Indeed, this observation otivate the use of a weighted haronic average. Assue that a has constant values a and a + to the left and right side of x, see Fig The corresponding values of the flux on either side of the discontinuity are: FLUX ¼a p p i x x i and FLUX þ ¼a þ p p iþ1=2 x iþ1 x ; where p is the pressure approxiation at the discontinuity. To enforce conservation of flux, we equate the two fluxes. Solving the resulting equation yields FLUX ¼FLUX þ ¼ p iþ1 p i Dx ð5þ ð6þ 1 x x i þ 1 1 x iþ1 x ; a Dx a þ Dx where Dx = x i+1 x i. Hence, a iþ1=2 ¼ 1 xx i þ 1 x iþ1 x 1. a Dx a þ Dx 3. In advance of the error analysis below, we note that if a is piecewise constant in one diension then and a1 have the sae average value, i.e. a1 ; 1 X ¼ a1 a1 ; 1 ¼ 0: X d Siilarly in two diensions, if a is piecewise constant and C d is a straight line, then we have another equality between the two diensional integrals of and with particular choice of quadratures (see below): 1 2 a1 a1 ; 1 X;M xt y þ 1 2 a1 a1 ; 1 X;M yt x ¼ 1 2 a1 a1 ; 1 X d ;M xt y þ 1 2 a1 a1 ; 1 X d ;M yt x ¼ 0: ð7þ 4. Because the variational forulation underlying the finite eleent ethod involves integrals over the volues, the finite eleent approxiation is infored by the behavior of the diffusion coefficient in the interior of the volue. Hence, the

6 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) finite eleent approxiation sees the geoetry of the diffusion interface inside each volue. This provides a consistent way to treat different interface geoetries. 3. Error analysis As discussed above, the purpose of the error analysis in this paper is to derive a coputational error estiate that produces an accurate estiate of the error in a quantity of interest coputed fro a particular nuerical solution. The estiate accurately quantifies the various contributions to the error, i.e. odeling, discretization, and quadrature. This provides the capability to decide how to iprove the discretization if the desired accuracy is not achieved. As we illustrate below with exaples, it is crucial to consider error in specific quantities of interest. This is sharply contrasted to the goal of a general a priori convergence analysis, which deterines the error in a nor. We present two error estiates. The first is an a priori analysis which bounds the effects of introducing a odel diffusion coefficient. This is an iportant coponent for the ain estiate, which is a goal-oriented a posteriori estiate on the error in a quantity of interest. The analysis for both results uses the finite eleent forulation of the ethod Analysis of the odeling error We decopose the error in p,h as e p ¼ p p ;h ¼ p p ;h þ ð p p Þ ¼ e p;h þ e p; : ð8þ The first error e p,h is the usual approxiation error due to nuerical discretization and as a consequence of our construction of the odeling proble can be analyzed as in [18]. The second error e p, estiates the difference between the analytic solutions of the original and odel proble. Likewise, we write: e u ¼ u u ;h ¼ u u ;h þ ð u u Þ ¼ e u;h þ e u; : ð9þ Theore 3.1. There is a constant C that depends on u and a, such that: 6 C ; ð10þ e u; and 6 C : e p; ð11þ The constant C requires an upper bound on the size of a, which does not exceed the size of a if the weighted haronic average is used to define a. The proof is given in Section A.1. The a priori bounds (10) and (11) show that the effects of error introduced by the odeling tend to zero as the esh is refined. It is possible to be ore precise in specific cases. For exaple, if C d is a straight line of length L and the diffusion is piecewise constant, then it follows that: 2 O L h j a a þ j 2 h 2 ¼ OðhÞ; ð12þ where h is the esh size and a and a + are the two values of a. Hence: ¼ O h 1=2 : ð13þ That is, the odeling error scales with the square root of the esh size A goal-oriented a posteriorierror representation We estiate the total error in a quantity of interest that can be expressed as a linear functional: Qðp; uþ ¼ðp; w p Þþðu; w u Þ; for w u 2 (L 2 (X)) 2 and w p 2 L 2 (X). For this purpose, we require two adjoint probles. The first is associated with the original proble: find / p 2 H 1 (X) and / u 2 H (div; X) such that: / u r/ p ¼ w u ; in X; r / u ¼ w p ; in X; h/ p ; v ni ¼0; on ox; 8 v 2 Hðdiv; XÞ: ð14þ The second is associated with the odel proble: find / p, 2 H 1 (X) and / u, 2 H (div;x) such that: / u; r/ p; ¼ w u ; in X; r / u; ¼ w p ; in X; h/ p; ;v ni ¼0; on ox; 8 v 2 Hðdiv; XÞ: We obtain: ð15þ Theore 3.2 (A posteriori error representation). The error in the quantity of interest is given by e p ; w p þ ð eu ; w u Þ ¼ u ;h; / u; P h / u; þ f ; / p; P h / p; hg; /u; P h / u; ni þ QE1 P h / u; þ QE2 Ph / p; u ; / u ; ð16þ where QE1ðvÞ ¼ þ a 1 u ;h;v hg; v niþhg;v ni M ; QE2ðwÞ ¼ðf ; wþðf ; wþ MxM y ; ux ;h ;vx T xm y þ uy ;h ;vy M xt y ð17þ ð18þ and P h and P h denote the lowest order Raviart Thoas projection and the usual L 2 projection, respectively. The proof is given in Section A.2. We define the Raviart Thoas projection in Section A.3. The first line of (16) estiates the contribution to the error given by the finite eleent discretization. The second line gives the contribution to the error fro using the quadrature that yields the finite volue schee. Together, the first and second lines give the contribution to the error fro discretization of the odel proble. The third ter ð Þu ; / u easures the contribution of odeling error arising because a is replaced by a. Note that this ter is not coputable and we deal with that below. Note that the odeling ter integrand is zero outside X d. The ter is large if both the odeling error and the adjoint solutions / p and / u are large in X d. The size of the adjoint solution depends strongly on the location of the support of the function deterining the quantity of interest. If we are interested in values far away fro the interface, the effect of the odeling on the error will be sall, see Fig Moreover, if the adjoint solution is nearly constant in X d, the approxiation property of the odel for the average value of the diffusion function (7) also leads to a sall odeling error contribution. We note that there are a nuber of possible finite eleent + quadrature discretizations that yield different finite volue schees. The first and second lines of (16) will vary according to the particular approach, for exaple avoiding the quadrature error expression [25,26] A coputable a posterioriestiate Two approxiations are required in (16) in order to obtain a coputable a posteriori estiate.

7 2774 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) Fig Left: We plot the data w that gives the average value in a sall region near (.8,.8) in the unit doain. Right: The corresponding adjoint solution for a proble with constant diffusion. Note that the adjoint solution is very near zero along the interface C d. The value of the solution at (.8,.8) will not be heavily affected by the odeling error. The first is coon for this approach, naely we nuerically solve the adjoint proble (15) to obtain approxiations U p, / p, and U u, / u,. As a consequence of Galerkin orthogonality, the approxiation of the adjoint proble cannot lie in the finite eleent space used for the priary proble, see [20,22,18]. Therefore, in order to solve the adjoint proble, we use the second order Raviart Thoas ixed finite eleent ethod on the sae esh used for the priary coputation. This insures that the ethod used to solve the adjoint proble involves the sae prograing structure as the finite volue schee for the forward proble. An alternative approach that is widely used is to use a finer esh for the adjoint solve, see [30]. Deterining the optial approach to solve the adjoint proble for discontinuous interface probles reains to be done. The second approxiation is needed to deal with the unknown odeling solution u and adjoint solution / u in the odeling ter ð Þu ; / u in (16). We write: u ; / u ¼ u;h ; / u; þ u;h ; / u; / u u u ;h ; /u ; where the first ter on the right is coputable. We have: u;h ; / u; 6 Ck ko h1=2 ; while u;h ; / u; / u u u ;h ; /u 6 C ax u u ;h ; /u; / u : The standard convergence estiate for the finite eleent discretization and the odeling estiate (10) applied to / suggest that the latter expression is higher order in h than the coputable odeling expression. In practice we neglect these ters when coputing the estiate. The resulting coputable a posteriori estiate is Theore 3.3 (Coputable A Posteriori error estiate). The error in the quantity of interest is approxiated by e p ; w p þ ð eu ; w u Þ u ;h; U u; P h U u; þ f ; U p; P h U p; hg; ð Uu; P h U u; Þni þ QE1ðP h U u; ÞþQE2 P h U p; u;h ; U u; : ð19þ 4. Nuerical results In this section, we present a nuber of exaples to illustrate aspects of the proposed ethod and error estiate The behavior of a as the esh is refined We test the properties of the odel with two probles. These exaples verify that the piecewise biquadratic approxiation a converges to a as the esh size decreases as deterined by (13) Exaple 1: discontinuity across a straight line The discontinuity interface C d is the line y x 0.73 = 0. The diffusion coefficient a equals 10 3 on one side of the line and 1 on the other. We plot for different esh sizes in Fig As expected the transition is sooth and the region X d decreases with the esh size Exaple 2: discontinuity across a circle The discontinuity interface C d is the circle (x 0.5) 2 + (y 0.5) 2 = while the diffusion coefficient a equals 10 3 inside the circle and 1 outside. Plots of at different esh sizes are given in Fig. 4.2.InFig. 4.2(b), we observe discontinuity in at several cell nodes. For this exaple, we relaxed the continuity condition at a few nodes between cut cells and their neighbors (during step 4 of Algorith 1) because the center of a neighboring cell on the diagonal actually sits on the interface C d Behavior of the error estiate and the contributions to the error We present three exaples to test the convergence rate of the cut-cell finite volue ethod, test the accuracy of the error estiate (19), and illustrate the behavior of the contributions to the error as the esh is refined. A standard way to easure the accuracy of an a posteriori error estiate is to use the effectivity index: ¼ Estiated Error : ð20þ Exact Error Ideally, the effectivity index should be one. We study the behavior of as the esh is refined. Note that the odeling proble changes as the esh is refined since the boundary region X d (on which a a) changes as the esh is refined Exaple 3: a one diensional exaple We solve the proble for which the diffusion coefficient is given by

8 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) Fig The diffusion is discontinuous across the line y x 0.73 = 0. We plot the odel diffusion for esh sizes 1/2, 1/8 and 1/32 left to right. Fig The diffusion is discontinuous across the circle (x 0.5) 2 +(y 0.5) 2 = We plot the odel diffusion for esh sizes 1/2, 1/8 and 1/32 left to right. ( aðxþ ¼ 103 ðx þ 1Þ; x > 0:45; x þ 1; x 6 0:45; with a discontinuity at x = We set the pressure function to be: ( pðxþ ¼ 103 sinðx 0:45Þþ2; x > 0:45; sinðx 0:45Þþ2; x 6 0:45; Table 4.1 Exaple 3: discretization, quadrature and odeling contributions to the error and effectivity indices. Grid level Discretization Quadrature Modeling (a) Quantity of interest at E E E E01 4.5E01 2.6E E02 6.5E02 9.5E E03 1.6E02 2.6E E03 3.6E03 1.9E E04 8.6E04 3.4E E05 2.2E04 1.8E06 (b) Quantity of interest at E E E E E E E01 9.5E01 3.4E E01 2.4E01 8.1E E02 6.1E02 2.3E E03 1.5E02 2.9E E03 3.8E03 4.7E07 and copute the corresponding righthand side f(x)=cos(x 0.45) + (x + 1) sin (x 0.45). For a quantity of interest, we take w u = 0 and w p = exp(100(x x 0 ) 2 ), giving the value of p in a sall region centered at a point x 0 2 [0,1]. Effectivity indices and different error contributions corresponding to x 0 = 0.45 and x 0 = 0.95 are listed in Table 4.1(a) and (b). We record the order of convergence in Table 4.2. We obtain accurate error estiates on all eshes that are at least odestly refined. Note that the residual and quadrature error contributions at 0.95 are larger than those at 0.45 as a consequence of the larger value of at 0.95 than at However, the odeling error contribution for an estiate of the value of the solution at 0.95 is less than that of the odeling error contribution for the value of the solution at This indicates that the jup in the Table 4.2 Exaple 3: errors and convergence rates. Grid level ke p k 1 ke u k 1 Error Order Error Order 1 5.4E E E E E E E E E E E E E E

9 2776 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) diffusion affects the error significantly if the quantity of interest involves values of the solution near the discontinuity Exaple 4: discontinuity across a straight line Next, we consider a two-diensional doain where the discontinuity interface C d is the line y x 0.73 = 0, with a =10 3 on one side of the line and a = 1 on the other. We set the true solution to be: Table 4.3 Exaple 4: discretization, quadrature and odeling contributions to the error and effectivity indices. Grid level Discretization Quadrature Modeling (a) Quantity of interest: average error over the whole doain E01 3.0E01 6.8E E04 3.3E02 2.0E E03 8.4E03 4.9E E04 1.9E03 1.3E E03 2.8E04 2.7E E06 1.4E05 1.2E E04 5.2E05 3.3E04 (b) Quantity of interest: average error over a sall region close to the discontinuity E02 4.6E02 9.8E E02 1.1E02 4.5E E03 3.6E03 3.2E E03 8.8E04 6.9E E03 6.8E05 1.2E E04 1.9E06 4.2E E04 1.9E05 1.4E04 (c) Quantity of interest: average error over a sall region reote fro the discontinuity E03 1.7E03 1.1E E03 1.4E03 9.1E E03 4.4E04 4.2E E04 1.1E04 3.0E E05 2.1E05 1.5E E06 5.9E06 7.7E E06 1.1E06 2.5E06 Table 4.4 Exaple 4: errors and convergence rates. Grid level ke p k 1 ke u k 1 Error Order Error Order 1 1.5E04 1.4E E E E E E E E E E E E E px; ð yþ ¼ exp ðy þ 0:45x 0:73Þ; >< y þ 0:45x 0:73 < 0; exp ðy þ 0:45x 0:73Þ1 þ 10 3 ; >: y þ 0:45x 0:73 P 0; and copute the corresponding reaction function f. We test the estiate using three different quantities of interest: the average error over the whole doain using w p =1 on [0,1] [0,1], the average error in a sall region close to the discontinuity using w p =1onjy x 0.73j and w 2 = 0 otherwise, the average error in a sall region far away fro the discontinuity using w p = 1 in (x 0.9) 2 +(y 0.9) and w p =0 otherwise, while w u 0 in all three cases. Effectivity indices and different error contributions are listed in Table 4.3(a) (c). We record the convergence rates in Table 4.4. We obtain accurate error estiates on all odestly refined eshes and better for all three quantities of interest. In the case of a quantity of interest localized away fro the discontinuous interface C d, we obtain accurate estiates on even crude discretizations. In all cases, the contribution to the error fro the odeling is about the sae size as the discretization errors. The contributions and the error are significantly saller for the quantity of interest that is localized away fro the discontinuous interface C d. We can explain this by considering the plots of the solutions of the adjoint probles corresponding to the three different quantities of interest, shown in Fig Reflecting the relative sizes of the contributions to the odeling error recorded in Tables 4.3(a) (c), we see that the effective support of the adjoint solution for the error in a sall region far fro the discontinuity is relatively localized, which eans that odeling has a relatively saller effect on this quantity of interest. On the other hand, the adjoint solution corresponding to the average error throughout the doain is significantly larger (as is its second derivative, which affects the discretization contribution) Exaple 5: Discontinuity across a circle We consider a discontinuity interface C d equal to the circle of radius 0.4 centered at (0.5,0.5) in the unit square [0,1] [0,1]. In the interior of the circle the diffusion coefficient is a =10 3 and on the exterior the diffusion coefficient is a = 1. We set the pressure to be: 8 expððx 0:5Þ 2 þðy0:5þ 2 0:16Þ; >< ðx 0:5Þ 2 þðy0:5þ 2 6 0:16; pðx; yþ ¼ 10 3 ðexpððx 0:5Þ 2 þðy0:5þ 2 0:16Þ1þ10 3 Þ; >: ðx 0:5Þ 2 þðy0:5þ 2 > 0:16; Fig Contour plots of the approxiate solutions to the adjoint probles (U p, ) in Exaple 4. Fro left to right we plot the adjoint solutions corresponding to the quantities of interest: the average error over the whole doain, over a sall region close to the discontinuity line and over a sall region far fro the discontinuity.

10 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) Table 4.5 Exaple 5: discretization, quadrature and odeling contributions to the error and effectivity indices. Level Discretization Quadrature Modeling (a) Quantity of interest: average error over the whole doain E09 2.0E E E E E E E E E E E E01 1.1E01 1.1E E E E E02 3.3E01 2.5E01 (b) Quantity of interest: average error over a sall region close to the discontinuity E09 1.3E E E E E E E E E E E E01 2.3E02 7.2E E E E E02 2.2E01 1.6E01 (c) Quantity of interest: average error over a sall region reote fro the discontinuity E10 2.8E E E E E E E E E03 3.1E01 2.2E E02 3.8E02 6.0E E03 3.6E02 2.5E E03 4.2E03 4.9E03 Table 4.6 Exaple 5: errors and convergence rates. Grid level ke p k 1 ke u k 1 Error Order Error Order 1 5.9E E E E E E E E E E E E E E and we copute the corresponding righthand side f. We test using the sae three quantities of interest used in Exaple 4. The effectivity indices and error contributions for the three different quantities of interest are listed in Table 4.5(a) (c), respectively. Convergence rates are provided in Table 4.6. The effectivity ratios are generally acceptable, though still varying significantly on discretization grids at a scale of Note that the odeling contributions is estiated to be zero at the initial esh level, since on the coarse initial esh the discontinuity lies copletely inside the (single) cell and is therefore invisible to the estiate. As before, the contributions to the error and the error are both significantly saller when the quantity of interest is localized away fro the discontinuous interface C d. In this exaple, we see that the contribution fro the odeling error is roughly the sae size as the contributions fro discretization Exaple 6: distribution of cell contributions to the error We illustrate the distribution of contributions to the error fro the cut-cells and cells in the rest of the doain. In particular, we exaine the effect of the curvature in C d on the contribution to the error. We choose C d to be a parabola as shown in the top in Fig The diffusion is 2 on one side of C d and 1 on the other. We estiate the error in the average value over the whole doain. We first consider a horizontal strip of cells which intersects the discontinuity in two places. The error contributions fro each cell are noralized and recorded in the top figure. As expected, the error contributions fro cut cells are significantly larger than the error contributions fro the cells away fro the discontinuity. In the reaining four subplots of Fig. 4.4, we plot the contributions of the cut-cells to the total error and to each of the error coponents. In particular, Figs. 4.4(a) (d) show the total error, the odeling error, the residual error, and the quadrature error along the interface respectively. In general, we expect a larger odeling error contribution in a region where the curvature of the discontinuity is larger and all contributions are soewhat larger near the region of highest curvature Exaple 7: the behavior of the estiates in a case with a coplex interface In this exaple, we study the behavior of the contributions to the error in a proble when the interface C d is rather coplicated and the forcing function is not anufactured to produce a given solution. The diffusion coefficient is 10 3 inside a cross shaped region in the iddle of the unit square and 1 outside. The righthand side f consists of a source odeled by a Gaussian function of height 100 near (1, 1) and a sink odeled by a Gaussian of height-100 near (0, 0). The boundary conditions are hoogenous Dirichlet. The quantity of interest is the average error in a sall region near the source. We consider both the case when sides of the cross are oriented parallel to the coordinate axis (though not aligned with cell boundaries) and the case when the cross is oriented at an angle with respect to the coordinate axes. The diffusion function for the two cases are plotted in Fig The nuerical solutions for the two cases on a grid are plotted in Fig The effect of the sudden increase in a can be seen clearly in the solutions. In Table 4.7, we record the nuerical error estiates and the estiated error contributions for a sequence of eshes. For sufficiently fine eshes, the error estiate and the contributions decrease roughly linearly in log of esh size. Refining the esh leads to a slow iproveent in overall accuracy in both cases. The case when the diffusion discontinuity is located along an interface that is oriented at an angle with the axes requires a finer esh before asyptotic behavior is seen. Note that at the coarsest esh level, the contribution to the odeling error is estiated as 0 because the discretization does not see the discontinuous behavior. The odeling error contributions appear to decrease roughly at a square root rate, ignoring the first two values. We plot this inforation in Fig Estiating the effect of error in the location of the interface We carry out a cell-wise a posteriori analysis to deterine the effects of error in the location of the interface, e.g. arising fro experiental easureent or nuerical coputation. We require the following two sallness assuptions to hold. 1. In X d, the intersection points with any line segents connecting the cell centers are in X d, see Fig Location errors do not ove the interface C d out of the sub-cells in which it is located in any one realization. As a special case, we can treat the situation in which the locations are known only for the intersection points with the line segents connecting cell centers through cell boundary centers, see Fig Note that this effectively sets a iniu cell size for a given set of location points. We suppose that the true location of the interface is described by a given set of locations with respect to cell centers {a i h,b i h,c i h} (see Fig. 5.1). The actual position is deterined by the easured

11 2778 D. Estep et al. / Coput. Methods Appl. Mech. Engrg. 200 (2011) Fig Error distribution test. (a) Cell contributions to the total error in a horizontal strip. The quantity of interest is the average error in the doain. (b) (e) Error contributions fro the cut-cells. (b) Contribution to the total error. Scale: 1.1E 6 to 1.6E 5. (c) Contribution to the odeling error. Scale: 1.4E 6 to 2.1E 6. (d) Contribution to the residual error. Scale: 5.3E 7 to 6.0E 6. (e) Contribution to the quadrature error. Scale: 1.9E 6 to 2.8E 5. Fig Diffusion coefficients and solutions on a grid for the two cases in Exaple 7. The diffusion has value a = 1000 inside the cross-shape region and a = 1 outside. ~i h; c ~ i h; b ~i hg. We let the corresponding odel or coputed values fa diffusion values be a and ã respectively. We note that a is deterined by the coefficients {c1,..., c9} with respect to the basis for the space of biquadratic functions through an equation of the for: 1 c1 B. C C A ¼ bðfai h; bi h; ci hgþ; c9 0 ð21þ