The aim of this work is to propose and analyze a new high order discontinuous Galerkin finite element method for the time integration of a Cauchy problem second order ordinary differential equations. These equations typically arise after space semi-discretization of second order hyperbolic-type differential problems, e.g., wave, elastodynamics and acoustics equation. After introducing the new method, we analyze its well-posedness and prove a-priori error estimates in a suitable (mesh-dependent) norm. Numerical results are also presented to verify the theoretical estimates. space-time finite elements, discontinuous Galerkin methods, second order hyperbolic equations
In this work we introduce a new two-level preconditioner for the ecient solution of large scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a "traditional" ne grid preconditioner, such as one-level Additive Schwarz, block Gauss-Seidel or block Jacobi preconditioners. The coarse component is built upon a new Multi Space Reduced Basis (MSRB) method that we introduce for the rst time in this paper, where a reduced basis space is built through the proper orthogonal decomposition (POD) algorithm at each step of the iterative method at hand, like the exible GMRES method. MSRB strategy consists in building reduced basis (RB) spaces that are wellsuited to perform a single iteration, by addressing the error components which have not been treated yet. The Krylov iterations employed to solve the resulting preconditioned system targets small tolerances with a very small iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner in dierent large scale computational settings related to parametrized advection diusion equations and compared with the current state of the art algebraic multigrid preconditioners.
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