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Abstract.In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numerical results are presented to assess our approach.Mathematics Subject Classification. 35J25, 35J60, 35K15, 35K55.

a b s t r a c tWe present a hybridizable discontinuous Galerkin method for the numerical solution of steady and time-dependent linear convection-diffusion equations. We devise the method as follows. First, we express the approximate scalar variable and corresponding flux within each element in terms of an approximate trace of the scalar variable along the element boundary. We then define a unique value for the approximate trace by enforcing the continuity of the normal component of the flux across the element boundary; a global equation system solely in terms of the approximate trace is thus obtained. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. If the problem is time-dependent, we discretize the time derivative by means of backward difference formulae. This results in efficient schemes capable of producing high-order accurate solutions in space and time. Indeed, when the time-marching method is ðp þ 1Þth order accurate and when polynomials of degree p P 0 are used to represent the scalar variable, the flux and the approximate trace, we observe that the approximations for the scalar variable, the flux and the trace of the scalar variable converge with the optimal order of p þ 1 in the L 2 -norm. Finally, we introduce a simple element-by-element postprocessing scheme to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous inter-element normal component, is shown to converge with order p þ 1 in the L 2 -norm. The new approximate scalar variable is shown to converge with order p þ 2 in the L 2 -norm. For the time-dependent case, the postprocessing does not need to be applied at each time-step but only at the times for which an enhanced solution is required. Moreover, the postprocessing procedure is less expensive than the solution procedure, since it is performed at the element level. Extensive numerical results are presented to demonstrate the convergence properties of the method.Published by Elsevier Inc.

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Abstract.In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree k for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of k + 1 in L 2 for any k ≥ 0. Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free, H(div)-conforming, and converges with order k + 2 for k ≥ 1 and with order 1 for k = 0. Numerical experiments are presented which validate the theoretical results.

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SUMMARYWe present an interpolation method for efficient approximation of parametrized functions. The method recognizes and exploits the low-dimensional manifold structure of the parametrized functions to provide good approximation. Basic ingredients include a specific problem-dependent basis set defining a lowdimensional representation of the parametrized functions, and a set of 'best interpolation points' capturing the spatial-parameter variation of the parametrized functions. The best interpolation points are defined as solution of a least-squares minimization problem which can be solved efficiently using standard optimization algorithms. The approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure. In addition, an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost. Numerical results are presented to demonstrate the accuracy and efficiency of the method.