R. A. Nicolaides. On the l 2 convergence of an algorithm for solving finite element equations. Math. Comp., 31:892--906, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....will allow, it is critical that the convergence rate of the method should be insensitive to the number of unknowns. The general solution strategy that appears most promising in this regard is multigrid, for which grid independent convergence rates have been proven for elliptic operators [1, 2, 3, 4, 5]. Although no rigorous extension of this theory has emerged for problems involving a hyperbolic component, methods based on multigrid have proven highly effective for inviscid calculations with the Euler equations [6, 7, 8] and remain the most attractive approach for Navier Stokes calculations ....

R.A. Nicholaides. On the l 2 convergence of an algorithm for solving finite element equations. Math. Comp., 31:892--906, 1977.

....one linear system to be solved on the finest mesh. Multilevel iteration is a general, powerful technique for solving nonlinear operator equations which can be approximated by an orderly sequence of discrete nonlinear systems. The linear multigrid schemes of Brandt [7] Hackbusch [9] Nicolaides [13], and possibly others, could be adapted in a similar manner to the one proposed here and would yield methods with similar properties. We have found our particular procedure to be effective on a variety on nonlinear PDE s; the implementation was a reasonably straightforward extension of the one ....

R. A. Nicolaides, On the ` 2 convergence of an algorithm for solving finite element systems, Math. Comp., 31 (1977), pp. 892--906.

....equations that arise from elliptic partial differential equations. While many convergence proofs already exist (e.g. Astrakhantsev [2] Bakhvalov [4] Bank and Dupont [5] Braess and Hackbusch [7] Douglas [10, 11] Federenko [12] Hackbusch [15, 17, 14, 16] Maitre and Musy [19] Nicolaides [20], Van Rosendale [21] Verfurth [23] Wesseling [24] and Yserentant [25] our assumptions and proof techniques are different and (we believe) enlightening. While we are principally interested in partial differential equations, our theory can be applied to symmetric, positive definite linear ....

....damping constant. The smoother has the property that solving the linear system BX = Y (where B, X , and Y are analogous to A, Z, and G) should be easy in comparison to solving AZ = G. In particular, the cost should be proportional to N j in order to obtain an optimal order work estimate [4, 5, 7, 10, 11, 12, 17, 20]. The accelerated case will be considered in Section 4. Consider the generalized eigenvalue problem a( k ; v) k b j ( k ; v) for all v 2 M j : Without loss of generality, we order the eigenvalues such that 0 1 2 : N j ) 1; and normalize k such that b j ( k ; i ) ffi ki and ....

R. A. Nicolaides, On the l 2 convergence of an algorithm for solving finite element equations, Math. Comp., 31 (1977), pp. 892--906.

....extensions and present some examples of classes of spaces to which the method can be successfully applied. Our two level scheme can be generalized to a k level scheme for k 2. However, the rate of convergence which our analysis would predict depends on N if k does. We, as well as several others [2, 4, 12, 14], have obtained for various k level schemes convergence results comparable to our two level scheme. These multi level schemes are relatively complicated, and the requirements of the elliptic equation and the space M are more severe; e.g. the requirement that all the meshes are quasi uniform. When ....

....of this argument will work for any fixed number of levels, one would like the number of levels to depend on N . In this case the above analysis will fail to show that the rate of convergence is bounded less than one independent of h. However, such results have been obtained for multilevel schemes [2, 4, 5, 12, 14]. To do so, the concept of simple block iteration has been abandoned in favor of recursively defined algorithms. Furthermore, all presently known proofs explicitly or implicitly require some elliptic regularity, that the meshes T h j all be quasi uniform, 9 and that the spaces M h j satisfy ....

R. A. Nicolaides, On the ` 2 convergence of an algorithm for solving finite element equations, Math. Comp., 31 (1977), pp. 892--906.

....constraints will allow, it is critical that the convergence rate of the method should be insensitive to the problem size. The general solution strategy that appears most promising in this regard is multigrid, for which grid independent convergence rates have been proven for elliptic operators [1, 2, 3, 4]. Although no rigorous extension of this theory has emerged for problems involving a hyper1 bolic component, methods based on multigrid have proven highly effective for inviscid calculations with the Euler equations [5, 6, 7] and remain the most attractive approach for Navier Stokes calculations ....

R.A. Nicholaides. On the l 2 convergence of an algorithm for solving finite element equations. Math. Comp., 31:892--906, 1977.

....analyzed, namely a W cycle with pre and post smoothing using damped Jacobi iterations for the Poisson problem on the unit square. A variable coefficient problem was analyzed by [7] structured triangle grids were considered in [2] and the multigrid method was further developed to general grids in [121]. The interest in multigrid methods was mainly theoretical at that time and was focused on the optimal complexity of the algorithms. Real applications were first considered by Brandt [42] who observed the computational efficiency of multigrid methods. He optimized the components of the algorithms ....

R. A. Nicolaides, On the l 2 convergence of an algorithm for solving finite element equations, Math. Comp., 31 (1977), pp. 892--906.

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R. A. Nicolaides. On the l 2 convergence of an algorithm for solving finite element equations. Math. Comp., 31:892--906, 1977.

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R. A. Nicolaides, On the l 2 convergence of an algorithm for solving finite element equations, Math. Comp. 31 (1977), 892 -- 906.

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