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Brandt, “Multi-level Adaptive Solutions to Boundary Value. Problems,” Math Comp., 31, , pp • Brandt, “ Guide to Multigrid Development, with.
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- Multigrid — MAESTROeX documentation
- Multigrid Methods
The beauty of multigrid methods comes from their simplicity and the fact that they integrate all of these ideas in such a way that overcomes limitations, producing an algorithm that is more powerful than the sum of its elements. This consent may be withdrawn. You can fix this by pressing 'F12' on your keyboard, Selecting 'Document Mode' and choosing 'standards' or the latest version listed if standards is not an option. North America. Log Out Log In Contact. OK Learn More. February 13, Get New Posts by Email.
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Leave a Comment Log In Registration. The solution of the problem 16 satisfies the following error estimates. In this section, for ease of notations we assume uniform polynomial order in each dimension, i. We only consider the non-trivial case, i. Invoking the non-negativity and the partition of unity properties of basis functions, we have. Using Sobolev inequalities, see  , and following the standard FEM approach, see e.
Then the following relation holds. We consider all levels of smoothness, i. For h -refinement knot insertion , we see that the extremal eigenvalues satisfy the theoretical estimates 29 for the discrete system of second order elliptic problems, i. As mentioned earlier, for reducing the smoothness we insert multiple knots. Smoothness from C 0 to C p Before proceeding further, we need to introduce some more notations which are needed for two- multi- grid analysis.
In terms of the operator A k , the discrete system 27 can be equivalently written as. To bound the spectral norm of the matrix A k we proceed as follows. For SPD matrices we know that the eigenvalues can be estimated in terms of the Rayleigh quotients. In this section we present a two-grid analysis for solving the linear system The purpose of this analysis is to show that the rate of convergence of the two-grid method for IGM is independent of the mesh-size h. In a two-grid method, the solution of the system 27 is first approximated on the fine grid using a simple stationary iterative method e.
Then, since on a coarser grid the smooth error can be well represented, and computations are cheaper, the resulting residual equation is transferred to the coarse grid and an error correction by solving the residual equation is computed. This error correction is then transferred back to the fine grid where it is added to the approximate solution obtained by the relaxation process. This is called the coarse-grid correction step. Finally, post-relaxation helps to further improve the fine-grid approximation by smoothing error components that may have been contaminated during the inter-grid transfer from the coarse to the fine grid.
The convergence rate of any two-grid method like this depends on the efficiency of the relaxation method smoother and on the approximation properties of the inter-grid transfer operators, and on how well smoothing and coarse-grid correction complement each other. For the two-grid analysis, we shall use the conventional notations h and H to denote the mesh size at the fine level and the coarse level, respectively.
Together with the space of basis functions V , the SPD operator A , and the linear functional f , these notations shall be used to reflect the mesh level. Let I h be the identity matrix and G h be the smoothing iteration matrix.
We know that the convergence of the two-grid method depends on the iteration matrix . In the following two sections we study the approximation property of the inter-grid transfer operators, and the smoothing property of the relaxation method. The h -independent convergence of the two-grid method, i.
To establish the approximation property we first prove the following Lemma, see e. For 42b we proceed as follows. Combining 42a and 42b we get.
Equivalently, albeit in a different terminology, see [14,26] for details, the estimate 43 reads. Illustration of the approximation property, i. In this section we recall the smoothing property of the symmetric Gauss—Seidel method. The symmetric Gauss—Seidel method 45 satisfies the smoothing property. Also, from [26, Lemma 6. Hence, using 52 we obtain.
As one might expect, the effect of one symmetric one forward followed by one backward Gauss—Seidel iteration is almost the same as for two forward Gauss—Seidel iterations. Due to this reason, we will use forward Gauss—Seidel iterations in our numerical tests for multigrid convergence. Illustration of the smoothing property, i. In this section we summarize some important consequences of the smoothing and approximation properties on the convergence of the classical multigrid algorithm in the setting of the isogeometric discretization.
Since the proofs of the quoted convergence results can be found in  , we confine ourselves to a short discussion without repeating any proofs. For convenience we first consider the symmetric case which is the simplest to analyze. Consider the iteration matrix 62 of the W -cycle method, i. Then the following convergence result holds true, cf. Next, consider the iteration matrix 62 of the V -cycle method, i. For the case of equal number of pre- and post-smoothing steps, i. The numerical results in the next section indicate, however, that these estimates are somewhat pessimistic, and that one obtains better convergence rates in practice.
The domain is chosen as a quarter annulus in the first Cartesian quadrant with inner radius 1 and outer radius 2, see . Furthermore, the operator P h H is chosen such that the coarse basis functions are exactly represented in the space of fine basis functions. At the finest level largest problem size , the parametric domain is divided into n 0 equal elements in each direction. The initial guess for iteratively solving the linear system of equations is chosen as a random vector.
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Let r 0 denote the initial residual vector and r it denote the residual vector at a given multigrid iteration n it. The following stopping criteria is used. In the tables, by M we collectively denote the multigrid method which is specified by the choice of the cycle, i.
For Example 1 , since the geometry mapping is identity, it suffices to choose the basis functions as B-splines. This number is sufficient since the Jacobian from the mapping is constant. We consider both the extreme cases of smoothness, namely, C 0 and C p We make the following observations. We now study the performance of V -cycle multigrid solver on a multi-patch geometry. The results presented in Table 9 show that the convergence behavior fits nicely between the convergence behavior for global C 0 and C p -1 smoothness, with a bias towards C p -1 smoothness.
We now consider Example 2 with variable coefficients. However, due to the exponential function, exact integration is not possible with respect to y -variable. We note that the results are qualitatively same as those with constant coefficients case see Table 8.
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We now consider Example 3 with curved boundary. The geometry for this example is represented by NURBS basis functions of order 1 in the radial direction and of order 2 in the angular direction, see . Note however that this is not detrimental to the optimality of any of the methods, which can be seen from the variable coefficients case presented in Table In Table 12 , we present the convergence factor and the number of iterations for V -, W -, and F -cycle multigrid methods. All the results are qualitatively similar to that of Example 1 with square domain.
Finally, we consider the three-dimensional problem described in Example 4. The results for V -cycle multigrid method are presented in Table 13 , which confirm the h -independence and optimality of the solver. As shown by the results of two-dimensional examples, the W - and F -cycle methods will not offer any improvement in convergence results, and are thus not repeated here.
For all the examples, we also tested the multigrid convergence for intermediate continuities C r , i. However, they are not reported here for brevity reasons. We also remark the following on the numerical results of high polynomial orders and where the exact solution has reduced regularity. It is known from finite elements literature that standard h -multigrid, which is the focus of this article, is not suited for high polynomial order. Most of the literature is for first and second order polynomials only.
Multigrid — MAESTROeX documentation
This fact is related to the smoothing properties of the classical smoothers like Jacobi, Gauss—Seidel or Richardson methods. These methods work effectively only when the error function is oscillatory, whereas the error function gets smoother with increasing polynomial order.
For high polynomial order, either p -multigrid should be used or different smoothers should be devised, both of which are beyond the scope of this article. In the presence of discontinuities in the coefficients, or due to the irregular geometry e. Secondly, the standard geometric multigrid methods are not tailored for such general problems and need special treatment.
The reduced regularity negatively affects the approximation property of Lemma 4 , and thus the overall convergence behavior of solver. Though specific problems can be treated to obtain optimal order convergence which involves more technical results , however, this is beyond the scope of this article. For such problems, the multi-patch techniques, such as the tearing and interconnecting approach of Kleiss et al.
We have presented multigrid methods, with V -, W - and F -cycles, for the linear system arising from the isogeometric discretization of the scalar second order elliptic problems. For a given polynomial order p , all multigrid cycles are of optimal complexity with respect to the mesh refinement. Despite that the condition number of the stiffness matrix grows very rapidly with the polynomial order, these excellent results exhibit the power of multigrid methods. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.
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