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A central question in numerical homogenization of partial differential equations with multiscale coefficients is the accurate computation of effective quantities, such as the homogenized coefficients. Computing homogenized coefficients requires solving local corrector problems followed by upscaling relevant local data. The most naive way of computing homogenized coefficients is by solving a local elliptic problem, which is known to suffer from the so-called resonance error dominating all other errors inherent in multiscale computations. A far more efficient modelling strategy, based on adding an exponential correction term to the standard local elliptic problem, has recently been proved to result in exponentially decaying error bounds with respect to the size of the local geometry. The ques- tions in relation with the accuracy and computational efficiency of this approach has been previously addressed in the context of periodic homogenization. The present article concerns the extension of mathematical and numerical study of this modified elliptic corrector problem to stochastic homog- enization problems. In particular, we assume a stationary, ergodic micro-structure and i) establish the well-posedness of the corrector equation, ii) analyse the bias (or the systematic error) originat- ing from additional exponential correction term in the model. Numerical results corroborating our theoretical findings are presented.
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how?' but also
why?', where?' and
what for?'.
The motivation for developing structure-preserving algorithms for special classes of problems originates independently in such diverse areas of research as astronomy, molecular dynamics, mechanics, control theory, theoretical physics and numerical analysis, with important contributions from other areas of both applied and pure mathematics. Moreover, it turns out that preservation of geometric properties of the flow not only produces an improved qualitative behaviour, but also allows for a significantly more accurate long-time integration than with general-purpose methods.
In addition to the construction of geometric integrators, an important aspect of geometric integration is the light it sheds on the relationship between geometric properties of a numerical method and favourable error propagation in long-time integration. A very successful organising principle is backward error analysis, whereby the numerical one-step map is interpreted as (almost) the flow of a modified differential equation. In this way, geometric properties of the numerical integrator translate seamlessly into structure preservation on the level of the modified equation. Much insight and rigourous error estimates over long time intervals can then be obtained by combining backward error analysis with the KAM theory and related perturbation theories for Hamiltonian and reversible systems. While this approach has been very successful for ordinary differential equations, much less is currently known about highly oscillatory systems and geometric integrators for partial differential equations.
Geometric numerical integration has been an active interdisciplinary research area since the last decade. Although the subject is in a lively phase of intensive development, the results so far are substantive and they impact on a wide range of application areas and on our understanding of core issues in computational mathematics. This is evidenced by the monographs \cite{HLW:GNI2002,LR:SMH2004}.design-through-analysis'' workflow. According to this paradigm, a prototype is first designed with Computer-aided-design (CAD) software and then finalized by simulating its physical behavior, which usually involves the simulation of Partial Differential Equations (PDEs) on the designed product. The simulation of PDEs is often performed via finite element discretization techniques.A severe bottleneck in the entire process is undoubtedly the interaction between the design and analysis phases. The prototyped geometries must undergo the time-consuming and human-involved meshing and feature removal processes to become
analysis-suitable''. This dissertation aims to develop and study numerical solvers for PDEs to improve the integration between numerical simulation and geometric modeling. The thesis is made of two parts. In the first one, we focus our attention on the analysis of isogeometric methods which are robust in geometries constructed using Boolean operations. We consider geometries obtained via trimming (or set difference) and union of multiple overlapping spline patches. As differential model problems, we consider both elliptic (the Poisson problem, in particular) and saddle point problems (the Stokes problem, in particular). As it is standard, the Nitsche method is used for the weak imposition of the essential boundary conditions and to weakly enforce the transmission conditions at the interfaces between the patches. After proving through well-constructed examples that the Nitsche method is not uniformly stable, we design a minimal stabilization technique based on a stabilized computation of normal fluxes (and on a simple modification of the pressure space in the case of the Stokes problem). The main core of this thesis is devoted to the derivation and rigorous mathematical analysis of a stabilization procedure to recover the well-posedness of the discretized problems independently of the geometric configuration in which the domain has been constructed. In the second part of the thesis, we consider a different approach. Instead of considering the underlying spline parameterization of the geometrical object, we immerse it in a much simpler and readily meshed domain. From the mathematical point of view, this approach is closely related to the isogeometric discretizations in trimmed domains treated in the first part. In this case, we consider the Raviart-Thomas finite element discretization of the Darcy flow. First, we analyze a Nitsche and a penalty method for the weak imposition of the essential boundary conditions on a boundary fitted mesh, a problem that was not studied before, not needed for our final goal, but still interesting by itself. Then, we consider the case of a general domain immersed in an underlying mesh unfitted with the boundary. We focus on the Nitsche method presented for the boundary fitted case and study its extension to the unfitted setting. We show that the so-called ghost penalty stabilization provides an effective solution to recover the well-posedness of the formulation and the well-conditioning of the resulting linear system.