A New Ensemble HDG Method for Parameterized Convection Diffusion PDEs

A new second order time stepping ensemble hybridizable discontinuous Galerkin method for parameterized convection diffusion PDEs with various initial and boundary conditions, body forces, and time depending coefficients is developed. For ensemble solutions in $L^∞$($0$, $T$; $L^2$($Ω$)), a superconvergent rate with respect to the freedom degree of the globally coupled unknowns for all the polynomials of degree $k$ ≥ $0$ is established. The results of numerical experiments are consistent with the theoretical findings.     

Four and more

We isolate several large classes of definable proper forcings and show how they include many partial orderings used in practice.     

A convergent multigrid cycle for the hybridized mixed method

We consider the application of a variable V‐cycle multigrid algorithm for the hybridized mixed method for second‐order elliptic boundary‐value problems. Our algorithm differs from the previous works on multigrid for the mixed method in that it is targeted at efficiently solving the matrix system for the Lagrange multiplier of the method. Since the mixed method is best implemented by first solving for the Lagrange multiplier and recovering the remaining unknowns locally, our algorithm is more useful in practice. The critical ingredient in the algorithm is a suitable intergrid transfer operator. We design such an operator and prove mesh‐independent convergence of the variable V‐cycle algorithm. Numerical experiments indicating the asymptotically optimal performance of our algorithm, as well as the failure of certain seemingly plausible intergrid transfer operators, are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

An a priori error analysis of an HDG method for an eddy current problem

This paper concerns itself with the development of an a priori error analysis of an eddy current problem when applying the well‐known hybridizable discontinuous Galerkin (HDG) method. Up to the authors' knowledge, this kind of theoretical result has not been proved for this kind of problems. We consider nontrivial domains and heterogeneous media which contain conductor and insulating materials. When dealing with these domains, it is necessary to impose the divergence‐free condition explicitly in the insulator, what is done by means of a suitable Lagrange multiplier in that material. In the end, we deduce an equivalent HDG formulation that includes as unknowns the tangential and normal trace of a vector field. This represents a reduction in the degrees of freedom when compares with the standard DG methods. For this scheme, we conduct a consistency and local conservative analysis as well as its unique solvability. After that, we introduce suitable projection operators that help us to deduce the expected a priori error estimate, which provides estimated rates of convergence when additional regularity on the exact solution is assumed.