Algebraic multigrid (AMG) is often an effective solver for symmetric positive definite (SPD) linear systems resulting from the discretization of general elliptic PDEs, or the spatial discretization of parabolic PDEs. However, convergence theory and most variations of AMG rely on A being SPD. Hyperbolic PDEs, which arise often in large-scale scientific simulations, remain a challenge for AMG, as well as other fast linear solvers, in part because the resulting linear systems are often highly nonsymmetric. Here, a novel convergence framework is developed for nonsymmetric, reduction-based AMG, and sufficient conditions derived for 2 -convergence of error and residual. In particular, classical multigrid approximation properties are connected with reduction-based measures to develop a robust framework for nonsymmetric, reduction-based AMG.Matrices with block-triangular structure are then recognized as being amenable to reductiontype algorithms, and a reduction-based AMG method is developed for upwind discretizations of hyperbolic PDEs, based on the concept of a Neumann approximation to ideal restriction (nAIR). nAIR can be seen as a variation of local AIR ( AIR) introduced in previous work, specifically targeting matrices with triangular structure. Although less versatile than AIR, setup times for nAIR can be substantially faster for problems with high connectivity. nAIR is shown to be an effective and scalable solver of steady state transport for discontinuous, upwind discretizations, with unstructured meshes, and up to 6th-order finite elements, offering a significant improvement over existing AMG methods. nAIR is also shown to be effective on several classes of "nearly triangular" matrices, resulting from curvilinear finite elements and artificial diffusion.
Summary In this paper, a few dual least‐squares finite element methods and their application to scalar linear hyperbolic problems are studied. The purpose is to obtain L2‐norm approximations on finite element spaces of the exact solutions to hyperbolic partial differential equations of interest. This is approached by approximating the generally infeasible quadratic minimization that defines the L2‐orthogonal projection of the exact solution, by feasible least‐squares principles using the ideas of the original scriptLscriptL∗ method proposed in the context of elliptic equations. All methods in this paper are founded upon and extend the scriptLscriptL∗ approach that is rather general and applicable beyond the setting of elliptic problems. Error bounds are shown that point to the factors affecting the convergence and provide conditions that guarantee optimal rates. Furthermore, the preconditioning of the resulting linear systems is discussed. Numerical results are provided to illustrate the behavior of the methods on common finite element spaces.
The effect of interpolated edges of curved boundaries on Raviart-Thomas finite element approximations is studied in this paper in the context of first-order system least squares methods. In particular, it is shown that an optimal order of convergence is achieved for lowestorder elements on a polygonal domain. This is illustrated numerically for an elliptic boundary value problem involving circular curves. The computational results also show that a polygonal approximation is not sufficient to achieve convergence of optimal order in the higher-order case. Introduction.Elliptic boundary value problems arising in science and engineering are often posed on domains with curved boundaries. For low-order finite elements it is usually sufficient to use a polygonal approximation of the boundary in order to ensure that the finite element order of convergence is not affected by the inaccurate representation of the boundary. For higher-order finite elements a more accurate representation of the boundary is required which is usually achieved by isoparametric finite elements; i.e., the same finite element space is used once more for the parametrization of a more accurate domain approximation. This is well-known in the case of standard nodal finite element spaces, and a complete theory can be found in many finite element books (see, e.g., [4,6]).Finite element methods which simultaneously approximate scalar (potential) and vector (flux) unknowns are also widely used due to the fact that improved approximation of fluxes is sometimes a desirable goal. Mixed and hybrid finite elements of saddle point structure [3] and first-order system least squares [2] are two popular approaches in this direction. Favorable conservation properties are among the advantages of the first of these classes of methods while the latter approach possesses an inherent error control and allows for simplified handling of coupling conditions. On the other hand, properties of either of these approaches are sometimes inherited by the other one due to closeness properties established in [5]. The treatment of curved boundaries in the context of edge-or face-based finite elements where Neumann boundary conditions are imposed on the normal flux is, however, rather seldom mentioned in the literature. A thorough analysis of the effect of polygonal boundary approximation on the Raviart-Thomas finite element approximations seems to be missing. The same appears to be true for the convergence analysis of parametric Raviart-Thomas elements in the higher-order case. General remarks in this direction can be found in the
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