A MULTIGRID ALGORITHM FOR IMMERSED INTERFACE PROBLEMS. Loyce Adams 1. University of Washington SUMMARY

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1 A MULTIGRID ALGORITHM FOR IMMERSED INTERFACE PROBLEMS Loyce Adams Dept. of Applied Mathematics University of Washington SUMMARY Many physical problems involve interior interfaces across which the coecients in the problem, the solution, its derivatives, the ux, or the source term may have jumps. These interior interfaces may or may not align with a underlying Cartesian grid. Zhilin Li, in his dissertation, showed how to discretize such elliptic problems using only a Cartesian grid and the known jump conditions to second order accuracy. In this paper, we describe how to apply the full multigrid algorithm in this context. In particular, the restriction, interpolation, and coarse grid problem will be described. Numerical results for several model problems are given to demonstrate that good rates can be obtained even when jumps in the coecients are large and do not align with the grid.. INTRODUCTION Many physical problems involve interior interfaces across which the coecients in the problem, the solution, its derivatives, the ux, or the source term may have jumps. These interior interfaces may or may not align with a underlying Cartesian grid. As an example, single phase Darcy ow in porous media is governed by the equation r (rp) = for the pressure p where = = with the permeability and the viscosity. If the medium has an interface across which the permeability varies, we know that [p] = and [p n ] = at this interface. Another example is Stokes ow where the interface is the boundary of a moving membrane or bubble, ([], [2]). A more complicated problem is to model the blood ow in the human heart. Here the interface is the boundary of the heart. Peskin [3] solves for the velocity of the uid in which the heart is immersed by solving the Navier-Stokes equations on a Cartesian grid with a delta function forcing term determined by the force the heart wall exerts on the uid. It can be shown [3] that this singular source term in the Navier-Stokes equations leads to jumps in pressure and the derivatives of velocity across the interface, and is discretized by discrete delta functions and transfered to the nearby Cartesian grid points. The velocity of the uid is then used to move the boundary of the heart to the next time. This procedure is called the immersed boundary method and seems to be only rst order accurate due to the way the force on the interface is spread to the Cartesian grid. Zhilin Li has recently developed an approach for discretizing elliptic problems with interior interfaces called the immersed interface method (IIM), ([4], [5]), which can handle both discontinuous coecients and singular sources. The idea is to compute on a Cartesian grid only, as in Peskin's This work was supported in part by the Scientic Computing Division of the National Center for Atmospheric Research, which is supported by NSF, and in part by Department of Energy grant DE-FG6-93ER258 and NSF grant DMS

2 2 immersed boundary method, but to nd accurate discretization stencils by incorporating knowledge of where the interface is located and the known jumps in the solution there, rather than by smearing the force with a discrete delta function. Li showed that second-order accurate discretizations could be found for a wide class of problems. Of course, there are problems of physical interest where the jumps at the interface are not known a priori and must be solved for rst before such an approach can be taken. Such problems and solution techniques are discussed in [6], but will not be the focus of this paper. The purpose of this paper is to describe how the full multigrid method (FMG) can be applied to the discrete equations that result from the IIM. Many authors have given ecient multigrid schemes for both symmetric and nonsymmetric systems of equations that arise from elliptic problems with discontinuous coecients. A partial list includes [7], [8], [9], [], [], [2], [3], and [4]. For problems with discontinuous coecients, care must be taken to devise a proper method of interpolation for the multigrid process. Much of the work done in this direction has assumed that any interfaces are aligned with the grid. However, Aaron Fogelson and James Keener have used multigrid schemes to solve non-aligned immersed interface problems for two-dimensional heat equations in regions with holes, and to solve for electrical potentials in cardiac tissues, [5]. One common approach is to use what is called operator-induced interpolation. That is, the stencil for the partial dierential equation incorporates information about the jumps in the coef- cients, and this stencil can be modied to produce a stencil for interpolation. Such an idea is found in [8] and [9] and has the advantage that explicit information about the interface need not be known directly. Black box multigrid can nd out from the problem stencil how to interpolate. This approach presumably can be used when the interface does not align with the grid, assuming the problem was discretized accurately. In the future, we plan to try this approach in conjunction with an IIM discretization. Here, we present a dierent approach. Since our stencil for the problem comes from the IIM, we have all the information about the interface and the jumps there. In this paper, we show how to use this information to build an O(h 2 ) accurate interpolation scheme. The results of this approach seem promising since V-cycle rates of.6 to.3 have been achieved. The paper is organized as follows. Section 2 gives an overview of the IIM. Section 3 describes our multigrid scheme with a derivation of the modied bilinear interpolation. Section 4 gives numerical results. Section 5 states the conclusions and avenues for further work. 2. IMMERSED INTERFACE METHOD OVERVIEW In this section, we review the immersed interface method. Details can be found in [4] and [5]. The IIM provides a discretization of elliptic PDEs that is O(h 2 ), where h is the uniform mesh spacing in both the x and y directions. Consider the problem () (u x ) x + (u y ) y = f(x; y) in [u]? = w(s)

3 3? Fig.. 6-pt Stencil for Irregular Points [u n ]? = v(s) where boundary conditions on are given and? is the interface, across which the jump in the solution and ux are assumed known as functions of the arc length s. The stencil for a regular point (all points of the standard 5-point stencil are on the same side of the interface) is the usual O(h 2 ) approximation that uses the 5-point stencil for u values and its edge midpoints for values. To discretize () at an irregular point, Li uses a sixth point stencil as shown in Figure where * represents a point (x ; y ) on the immersed interface and looks for a formula at the center point of the form (2) (u x ) x + (u y ) y = 6X i u i? c + O(h) where u i denotes the i-th point in the 6-point stencil, the i 's are the coecients to be determined, and c is a correction term that can be computed once the i 's are known. Requiring the truncation error in (2) to be O(h) at the irregular points and the truncation error to be O(h 2 ) at the regular points is sucient to guarantee that a global error of O(h 2 ) is achieved everywhere. Let and be the normal and tangential directions at the point (x ; y ) which are given by (3) = (x? x ) cos + (y? y ) sin =?(x? x ) sin + (y? y ) cos : We then expand u i about the point (x ; y ) on the interface after changing to the (; ) variables. That is, (4) u i = u + i u + i u + i i u i u i u + ::: where * means to take the + or - limiting value on the outside or inside of the interface, respectively. Then we have 2 unknown terms on the right hand side in (2) and 6 unknown i 's. But, since we know the jumps from (), the following jump conditions can be derived for the special case where = in inside the interface and = out outside.

4 4. u + = u? + w 2. u + = u? + w 3. u + = u? + v= out 4. u + = u? + (? ) u? + w + v = out 5. u + = u? + (? ) u? + w? v= out 6. u + = u? + (? ) u? + (? )u? + v= out? w + [f]= out The variables w and v are functions of only, = in = out, the interface is described parametrically as = (), and all variables in the conditions above have been evaluated at (; ) = (; ) which corresponds to the * point on the interface. Next, we substitute these six conditions into (4) and then substitute (4) into (2) to get six equations in six unknowns for the i 's. Once these are found, c is determined from the i 's and the jump conditions. The end result gives an O(h 2 ) approximation to the exact solution u that satises (u xx + u yy ) = f. To use the IMM to generate the problem, the user must specify w, v, and [f] at control points (X; Y ) along the interface. The program ts a cubic spline through X, Y, v, w, and [f] at these control points to dene X(s), Y (s), v(s), w(s), and [f](s) as functions of the arc length parameter s. The quantitites in the jump relations are then derivable from these functions. As part of the procedure, each grid point is typed as being inside, outside, or on the interface, as well as being regular or irregular. One advantage of this approach is that the same interface can be used on each grid of a multigrid routine. That is, as we rene the grid, we need not rene a grid representing the interface. It is sucient to specify a relatively small number of control points, depending on the smoothness of the interface, in order to describe the interface with a spline. Of course, this procedure can not handle problems with interfaces that can not be well represented with a cubic spline. A future improvement to the implementation of the IIM would be to describe the immersed interface with a level set formulation. The coding involved would be reduced signicantly and we plan to do this before we tackle problems with multiple interfaces. 3. A FULL MULTIGRID SCHEME The result of the IIM is a discrete system of equations, A h u h = f h, on the nest grid with uniform mesh spacing h in each coordinate direction. The goal is to develop a multigrid strategy to solve this system quickly. Unlike the Black Box multigrid approach of [8] and [9] which uses operator-induced interpolation, we base our strategy on knowledge of where the interface is located and the jumps there. We have not yet compared our approach to Dendy's but we can claim to get fairly good multigrid rates with our approach for this class of problems. The basic components of full multigrid are the smoother, the restriction operator, the interpolation operator, and the coarse grid problem. We now describe what we choose for each. For all our test cases, point-rowwise Gauss-Seidel worked ne as the smoother. More complicated problems with larger jumps in the coecients may require a more sophsicated smoother. The coarse grid problem, A 2h u 2h = f 2h was taken to be the output of the IIM method with mesh size 2h. This

5 5? u 3 u 4 u u u 2 Fig. 2. Interpolation for u choice seems to limit the size of the coarsest grid to be (h = :4) for problems with ratios in = out = 2. It is possible to dene A 2h to satisfy the Galerkin condition, A 2h = I 2h h A h I2h, h but this has not been implemented yet. With the exception of the limitation in grid size described above, our denition of the coarse grid problem worked ne. The interpolation operator we used is a modied bilinear interpolation in the, coordinates for grid cells that contain an interface. If the cell does not contain an interface, the interpolation reduces to ordinary bilinear interpolation. To interpolate to the ne grid point at the center of a coarse grid cell, we build a formula based on the corner values of this cell plus a correction term. To interpolate to the midpoint of a vertical(horizontal) edge we choose either the cell to the east or west (or north or south) and nd a formula based on the corner values of this chosen cell plus a correction term. The cell choice depends on the location of the interface relative to the ne grid point for which we are seeking an interpolated value. For example, if one cell is regular (no interfaces crossing its boundary) it is preferred over the irregular cell. If both cells are irregular, an attempt is made to choose the one that will produce the most accurate value. To describe the scheme mathematically, we consider the chosen coarse grid cell shown in Figure 2 where u i are the four coarse grid values, u is a ne grid point whose value we wish to nd, and is a point (x ; y ) on an interface cutting through the cell. During the continuation phase of FMG, we look for a solution to u of the form (5) u = 4X i u i? c much in the same way as the 6-point stencil was found for the PDE in the IIM method. Again, let i and i be the transformed variables given in (3) of the previous section, and expand each u i and u about (x ; y ) on the interface using (4). Using the jump conditions given for the IIM method, we get the system A = b for the i 's after equating the coecients of u?, u?, u?, and u?. The matrices A and b are given below,

6 6 (6) A = (7) b = and the correction term c = c? c where (8) c = 4X i i (w + i v= out + i w + i i (w + v = out ) ) c = (w + v= out + w + (w + v = out ) ) : In the above equations, if the cell is regular, i = and =. If the cell is irregular, i = if the point (x i ; y i ) is outside the interface and i = if it is inside or on the interface. Likewise, if the cell is irregular, = if the point (x; y) is outside the interface and = if it is inside or on the interface. If the point (x i ; y i ) is outside the interface then i = in = out and i = if it is inside or on the interface. Likewise, = in = out if the point (x; y) is outside the interface and = if it is inside or on the interface. Also i = (? i ), and = (? ). Upon examination of these equations, it can be seen that for each irregular coarse grid cell, we really are calculating two bilinear functions, each of the form u = a + b + c + d. Each function interpolates the coarse grid points on the respective side of the interface (4 conditions). In addition, the functions are such that the jump conditions [u]?, [u ]?, [u ]?, and [u ]? are satised at the interface point (x ; y ) (the remaining four conditions). Since the terms left o are O(h 2 ), the formula is O(h 2 ) for u, relative to the true solution of the partial dierential equation. Hence, these formulas should give good results if the second derivatives are not too large relative to the mesh spacing. During the V-cycle, we need to interpolate the error e 2h to the ner grid. The same approach could be used if we knew [e 2h ]? and [e 2h n ]? at the interface. These are not known, but if the smoother is doing a good job, it makes sense to set these jumps to zero. Then the same i 's that were calculated during the continuation phase for interpolating u are the proper values to use for interpolating the error as well. This approach works well in practice as seen in the results in the next section. We choose the restriction operator to be a multiple of the transpose of the interpolation operator just described. In particular, I 2h h = :25(I2h) h T. In the case of regular cells, this reduces to full

7 weighting. For irregular cells, the stencil has a width of two grid cells in each direction, excluding other coarse grid points. The data structure used is a 55 stencil with other coarse grid connections set to zero. 4. NUMERICAL RESULTS Several test problems were run using the full multigrid scheme described above. For each test problem, we use the notation V(a,b) to denote that a pre- and b post- smooths were used in each V- cycle. More cycles than necessary to reach truncation error were taken for the purpose of studying the convergence to the solution of the discrete system. In all problems, about 3 V-cycles were sucient to reach truncation level. In each Table, derr denotes the dierence between the computed solution and the exact solution of the dierence equations and res is the residual. The grid size given for each Table is that of the nest grid. In the Figures, err is the dierence between the computed solution and the exact solution of the partial dierential equation. Problem. 7 is The domain is the (?2; 2) (?2; 2) square and the interface? is the unit circle. The problem (u xx + u yy ) = f in = :5, out = :, f in = 2:, f out = u in = x 2 + y 2, u out = x 2? y 2 [u]? =?2y 2 [u n ]? = 2 out (x 2? y2 )? 2 in Table shows rates of each V-cycle to be.3 for both the discrete error and the residual for a 2-level scheme on a 4 4 grid with 2-pre and 2-post smooths. Notice that the modied bilinear interpolation used in continuation gave a starting guess on the nest grid of.2. This is good since the mesh size on the nest grid is h = :. Cycle jjderrjj jjresjj rate derr rate res Starting :2? :5 4 -V :23?2 : V :26?3 : V :34?4 : V :44?5 :96? V :56?6 :2? V :73?7 :6? V :94?8 :2?3.3.3 Table. Problem : V(2,2), 4x4, 2-levels Table 2 gives 2-level V(4,4) results for Problem. Notice that the rates went down from.3 to.6.

8 8 Cycle jjderrjj jjresjj rate derr rate res Starting :2? :5 4 -V :37?3 : V :25?4 : V :4?5 :2? V :93?7 :3? V :6?8 :83? V :39?9 :53? V :25? :34?6.6.6 Table 2. Problem : V(4,4), 4x4, 2-levels Table 3 gives 3-level results for an 8 8 ne grid and V(4,4). Notice that we still get rates of.6 with 3-levels. Also note that even though the level 2 problem was solved with only V-cycle, the starting error for level 3 was.6. Cycle jjderrjj jjresjj rate derr rate res Starting :6? :5 5 -V :96?3 : V :5?4 : V :78?6 :64? V :5?7 :38? V :28?8 :4? V :9?9 :9?5.7.6 Table 3. Problem : V(4,4), 8x8, 3-levels Figure 3 shows the computed solution for this problem with V(4,4) for an 8 8 ne grid after 7 V-cycles. Notice the sharpness of the jump at the interface. Figure 4 shows the associated err. This error, O(?5 ), is concentrated along the interface as expected since the truncation error is largest there. We note that the discrete error, derr, is O(? ) and is much smoother at the interface due to the multigrid smoothing. PROBLEM 2. The problem domain is the (?2; 2) (?2; 2) square and the interface? is the unit circle. The problem is (u xx + u yy ) = f in =, out =, f in = f out = 2 u in = x 2, u out = x 2 [u]? =?999(x 2 )

9 Fig. 3. u for Problem. x Fig. 4. err for Problem. [u n ]? =, [u n ]? =?999(2x 2 ) Note that this problem has a jump in the normal derivative at the interface even though the jump in the ux is zero. Table 4 shows rates for a 2-level method with V(4,4) to be.3 for the discrete error and.6 for the residual.

10 Fig. 5. Starting err for Problem 2. Cycle jjderrjj jjresjj rate derr rate res Starting : 2 :2 5 -V :9 : V :48?2 : V :2?3 :2? V :36?5 :74? V :?6 :49? V :3?8 :3? V :?9 :2?6.4.6 Table 4. Problem 2: V(4,4), 4x4, 2-levels Of special note in Table 4 is the starting error produced by the modied bilinear interpolation during continuation. At rst sight this error of looks quite bad. But notice that u xx = 2 for points inside the interface, and the term 2 (x? x ) 2 u xx that is not included in the bilinear interpolation is exactly. In fact, Figure 5 shows the starting error to be very sharp at the interface, reecting the fact that the truncation error has a dierent constant for points inside and outside the interface. This is the best we can hope to accomplish with bilinear interpolation for this problem. We do not plot the solution and error for this problem since the graphs are quite similar to Problem in that the jumps are captured very sharply.

11 Problem 3. As an application we consider single-phase saturated ow governed by Darcy's law, (9) ~u =?rp r ~u = where ~u = (u; v) T is the velocity vector and = = with a discontinuous permeability at the interface. Such problems arise in groundwater ow and contaminant transport. Combining the equations above we get the elliptic equation () r (rp) = [p]? = [p n ]? = for the pressure p. Equation () is then discretized with the IIM and solved using multigrid. The velocities of the ow are then determined from (9). A similar strategy that was used for modied bilinear interpolation can be used to devise an O(h) formula for p x and p y in cells with interfaces. One could also get O(h) formulas by using onesided dierences on the correct side of the interface. If the pressures, p, are calculated by multigrid on a grid of size h, modied bilinear interpolation can be used to give p at cell-centers and edges on a grid of size h=2. Then the needed information is available to nd derivatives to O(h) at grid points of the h=2 grid. A more exact, though more expensive method, is to calculate pressures on a grid of size h=2 for use in the derivative calculation on a grid of size h. This was the approach that was taken in the results that follow. Once derivatives are found, we solve the equation () q t + ~u rq = for advection of a contaminant with concentration q. This is done with LeVeque's Clawpack software on a uniform grid, ([6], [7]). For the test problem we take to be the (?2; 2) (?2; 2) square and? to be the interface shown in Figure 6. On the square, p = at the left boundary, p = at the right boundary, and p y = at the top and bottom boundaries. The permeabilities are = 5 inside the interface and = outside the interface. Initially, q = and at the left(inow) boundary q =, and an 8 8 computational grid is used. Figure 6 shows the velocities that were determined by dierencing the pressure that came out of the multigrid routine. Since is larger inside the interface, the velocity should move the

12 2 q at time t = Fig. 6. Velocities for Problem 3. q at time t =.5 q at time t = q at time t =.5 q at time t = 2 Fig. 7. Contours of q for Problem 3. contaminant quicker through this region than around it. This is what is observed at four times as shown in Figure 7. Our approach did give sharp results for the moving front of the contaminant even though the Clawpack routine used did not have knowledge of where the interface was located. 5. CONCLUSIONS We have demonstrated that a full multigrid algorithm can be designed for interface problems where the jumps in coecients, solution, derivatives, ux, or source term are not aligned with the underlying Cartesian grid. This algorithm correctly solves the ne grid problem generated by Li's

13 3 IIM and hence gives a second-order accurate solution to the partial dierential equation. The multigrid solution for Problem with jumps in the coecients, the solution, the ux, and the source term, was obtained at a rate of.3 using V(2,2) with 2 levels,.6 using V(4,4) with 2-levels, and.6 using V(4,4) with 3-levels. For Problem 2, with a large jump in [u n ], but [u n ] =, we obtained rates of.3 for errors and.6 for residuals using V(4,4) and 2-levels. In order to achieve such rates, a modied bilinear interpolation scheme that takes advantage of known jumps in the problem at the interface as well as knowledge of where the interface is located was developed. If the second derivatives in u (for continuation) or discrete error (for V- cycle) are not too big, this interpolation can be expected to give good results to O(h 2 ). If a coarse grid cell is regular, then the modied interpolation reduces to ordinary bilinear interpolation, and restriction becomes full-weighting. For V-cycle interpolation, the assumption that [e]? = and [e n ]? = seems to be a reasonable one since we achieved a factor of 7 to improvement over the pre-smoothed result after doing coarse grid correction. This multigrid approach was used successfully to generate pressures from which velocities were obtained for the groundwater ow application in Problem 3. The contaminant was advected in this velocity eld using a Clawpack routine that did not know about the location of the interface. Results showed that the contaminant front was very sharp. There are still many improvements that can be made or questions that should be answered. First, the coarse grid problems come directly from an immersed interface formulation on the given grid level, not from a Galerkin condition of the ne grid problem. It is possible that one could use even coarser grids if a Galerkin approach is used. In addition, a Galerkin formulation may be more amenable to dierent smoothing strategies than our approach and could be benecial when more complicated problems are tackled. Second, we plan to compare this approach to the operator-induced interpolation approaches that others have taken. In particular, Dendy's Black Box solver for nonsymmetric problems, [9], could take the 6-point stencil generated by the IIM and infer an interpolation strategy, as well as automatically determining the coarser grids without explicit knowledge of the interface. ACKNOWLEDGMENTS The author would like to thank Randy LeVeque for useful discussions about the immersed interface method and for running the Clawpack routine for Problem 3. Steve McCormick and Tom Manteuel have provided a stimulating environment at the Applied Math Program, University of Colorado, Boulder for learning about multigrid. Joel Dendy and Victor Brandy gave tremendous insight into multigrid approaches for problems with discontinuities. Thanks also to Paul Swartztrauber in the Scientic Computing Division at the National Center for Atmospheric Research for arranging nancial support during my sabbatical year.

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