Adaptive Petrov--Galerkin Methods for First Order Transport Equations

Abstract

Abstract. We propose a general framework for well posed variational formulations of linear unsymmetric operators, taking first order transport and evolution equations in bounded domains as primary orientation. We outline a general variational framework for stable discretizations of boundary value problems for these operators. To adaptively resolve anisotropic solution features such as propagating singularities the variational formulations should allow one, in particular, to employ as trial spaces directional representation systems. Since such systems are known to be stable in L 2 special emphasis is placed on L 2 -stable formulations. The proposed stability concept is based on perturbations of certain "ideal" test spaces in Petrov-Galerkin formulations. We develop a general strategy for realizing corresponding schemes without actually computing excessively expensive test basis functions. Moreover, we develop adaptive solution concepts with provable error reduction. The results are illustrated by first numerical experiments.

Adaptivity and variational stabilization for convection-diffusion equations

“…We conclude this section with the simple observation that the above strategy of first prescribing Y and then choosing X through setting v X := Av Y can be reversed which is the point of view taken in [9] for a different problem class. In fact, first prescribing X and setting y Y = A * y X yields the dual space f Y = A −1 f X so that one obtains again the desired mapping property Au Y = u X and (X, Y )-stability with perfect condition κ X,Y (A) = 1.…”

“…Here we only give a short motivation and refer to [9] for a detailed discussion. A first natural weak formulation is obtained by integrating the reduced equation b · ∇u + cu = f multiplyed by a test function which yields…”

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

“…Stability factors. If the (Navier)-Stokes operator is parameter-dependent, so is the lower bound of the stability factor (38) or (43). In this case, computing its lower bound according to a suitable Offline/Online splitting is not easy.…”

“…In this case, computing its lower bound according to a suitable Offline/Online splitting is not easy. We face it by using the so-called Successive Constraint Method (SCM) 4 which converts the eigenproblem corresponding to the computation of (38) or (43) on the successive solution of suitable linear optimization problems. This algorithm has been applied for the first time to saddle point Stokes problems in [106], while a first extension to the nonlinear Navier-Stokes case has been considered in [87].…”

“…However, one is then left with the question of how to choose the test space. Recent works on optimal or near-optimal choice of Petrov-Galerkin test spaces were presented in [40,41] and [38]. These options are "optimal" in the sense that they give the best possible ratio of continuity constant to stability constant in the energy norm estimates.…”