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Methods for solving discontinuous Galerkin formulations of the Poisson equation by coupling p multigrid to geometric multigrid are investigated. The simple approach of performing iterative relaxation on solution approximations of decreasing polynomial degree p down to p = 0 and then applying geometric multigrid is ineffective. The transition from p = 1 to p = 0 causes the performance of the entire iteration to degrade because the long wavelength eigenfunctions of thep = 1 discontinuous system are not represented well in the p = 0 space. A new approach is proposed that coarsens from thep = 1 discontinuous space to thep = 1 continuous space. This approach eliminates the problems caused by the p = 1 to p = 0 transition. Furthermore, the p = 1 continuous space is a standard finite element space for which geometric multigrid is well-defined. In addition, when the discontinuous Galerkin equations are restricted to a continuous space, one recovers the Galerkin formulation of continuous finite elements. Thus, applying geometric multigrid to this system is straightforward and effective. Numerical tests agree well with the analysis and confirm that the new approach gives rapid convergence rates that are grid-independent and only weakly sensitive to the polynomial order.

Ap-multigrid (wherep is the polynomial degree) discontinuous Galerkin method is presented for the solution of the compressible Euler equations on unstructured grids. The method operates on a sequence of solution approximations of different polynomial orders. A distinct feature of this p-multigrid method is the application of an explicit smoother on the higher-level approximations (p > 0) and an implicit smoother on the lowest-level approximation (p = 0), resulting in a fast (and low) storage method that can be efficiently used to accelerate the convergence to a steady-state solution. Furthermore, this p-multigrid method can be naturally applied to compute the flows with discontinuities, in which a monotonic limiting procedure is usually required for discontinuous Galerkin methods. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing slip boundary conditions for curved geometries [Krivodonova, L., and Berger, M., 'High-Order Implementation of Solid Wall Boundary Conditions in Curved Geometries,' Journal of Computational Physics, Vol. 211, No. 2, 2006, pp. 492-512; and Luo, H., Baum, J. D., and Lohner, R., 'On the Computation of Steady-State Compressible Flows Using a Discontinuous Gaierkin Method,' International Journal for Numerical Methods in Engineering, Vol. 73, No. 5, 2008, pp. 597-623]. A variety of compressible flow problems for a wide range of flow conditions from low Mach number to supersonic in both two-dimensional and three-dimensional configurations are computed to demonstrate the performance of this p-multigrid method. The numerical results obtained strongly indicate the order-independent property of this p-multigrid method and demonstrate that this method is orders-of-magnitude faster than its explicit counterpart. The performance comparison with a finite volume method shows that using this p-multigrid method, the discontinuous Galerkin method, provides a viable, attractive, competitive, and probably even superior alternative to the finite volume method for computing compressible flows at all speeds.

A constructive procedure for factoring a polynomial paraunitary matrix as a product of atomic paraunitary matrices is considered. The generic form of the paraunitary factors results from the use of projection matrices. In the univariate case, the factorization carried out in the z-transform domain collapses to other univariate paraunitary matrix factorization methods. In the multivariate case, the generic factor is constructed based on the assumption of its existence. Nontrivial examples for multiband univariate, bivariate, and trivariate cases are given to illustrate the simple and straightforward algorithmic implementation. Index Terms--Multivariate polynomial paraunitary matrix factorization, projection operators.

Multidimensional (n-D) polynomial matrix factorizations are intimately linked to many problems of multidimensional systems and Signal processing. This paper gives a new result for a n-D polynomial matrix to have an minor prime factorization using methods from computer algebra. This result may be regarded as a generalization of a previous criterion under a special restriction [IEEE Trans. Circuits Syst. II: Exp Briefs, vol. 52, no. 9 (2005)]. Examples are given to illustrate results using computer algebra software system Singular. Index Terms--Matrix factorizations, minor primeness, multidimensional (n-D) systems, n-D polynomial matrices, polynomial ring.