On Multiscale Methods in Petrov-Galerkin formulation

In this work we investigate the advantages of multiscale methods in Petrov-Galerkin (PG) formulation in a general framework. The framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space{, which only contains negligible fine scale information}. The multiscale space can then be used to obtain accurate Galerkin approximations. As a model problem we consider the Poisson equation. We prove that a Petrov-Galerkin formulation does not suffer from a significant loss of accuracy, and still preserve the convergence order of the original multiscale method. We also prove inf-sup stability of a PG Continuous and a Discontinuous Galerkin Finite Element multiscale method. Furthermore, we demonstrate that the Petrov-Galerkin method can decrease the computational complexity significantly, allowing for more efficient solution algorithms. As another application of the framework, we show how the Petrov-Galerkin framework can be used to construct a locally mass conservative solver for two-phase flow simulation that employs the Buckley-Leverett equation. To achieve this, we couple a PG Discontinuous Galerkin Finite Element method with an upwind scheme for a hyperbolic conservation law

Convergence of a discontinuous Galerkin multiscale method

A convergence result for a discontinuous Galerkin multiscale method for a second order elliptic problem is presented. We consider a heterogeneous and highly varying diffusion coefficient in $L^\infty(\Omega,\mathbb{R}^{d\times d}_{sym})$ with uniform spectral bounds and without any assumption on scale separation or periodicity. The multiscale method uses a corrected basis that is computed on patches/subdomains. The error, due to truncation of corrected basis, decreases exponentially with the size of the patches. Hence, to achieve an algebraic convergence rate of the multiscale solution on a uniform mesh with mesh size $H$ to a reference solution, it is sufficient to choose the patch sizes as $\mathcal{O}(H|\log(H^{-1})|)$. We also discuss a way to further localize the corrected basis to element-wise support leading to a slight increase of the dimension of the space. Improved convergence rate can be achieved depending on the piecewise regularity of the forcing function. Linear convergence in energy norm and quadratic convergence in $L^2$-norm is obtained independently of the forcing function. A series of numerical experiments confirms the theoretical rates of convergence

Multiscale methods for problems with complex geometry

We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments

Hybridized CutFEM for Elliptic Interface Problems

We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach. We discuss how optimal error estimates for the method are obtained using the tools of cut finite element methods and prove a condition number estimate for the Schur complement. Finally, we present illustrating numerical examples

Adaptive multilevel subset simulation with selective refinement

In this work we propose an adaptive multilevel version of subset simulation to estimate the probability of rare events for complex physical systems. Given a sequence of nested failure domains of increasing size, the rare event probability is expressed as a product of conditional probabilities. The proposed new estimator uses different model resolutions and varying numbers of samples across the hierarchy of nested failure sets. In order to dramatically reduce the computational cost, we construct the intermediate failure sets such that only a small number of expensive high-resolution model evaluations are needed, whilst the majority of samples can be taken from inexpensive low-resolution simulations. A key idea in our new estimator is the use of a posteriori error estimators combined with a selective mesh refinement strategy to guarantee the critical subset property that may be violated when changing model resolution from one failure set to the next. The efficiency gains and the statistical properties of the estimator are investigated both theoretically via shaking transformations, as well as numerically. On a model problem from subsurface flow, the new multilevel estimator achieves gains of more than a factor 200 over standard subset simulation for a practically relevant relative error of 25%

Multiscale Methods and Uncertainty Quantification

In this thesis we consider two great challenges in computer simulations of partial differential equations: multiscale data, varying over multiple scales in space and time, and data uncertainty, due to lack of or inexact measurements. We develop a multiscale method based on a coarse scale correction, using localized fine scale computations. We prove that the error in the solution produced by the multiscale method decays independently of the fine scale variation in the data or the computational domain. We consider the following aspects of multiscale methods: continuous and discontinuous underlying numerical methods, adaptivity, convection-diffusion problems, Petrov-Galerkin formulation, and complex geometries. For uncertainty quantification problems we consider the estimation of p-quantiles and failure probability. We use spatial a posteriori error estimates to develop and improve variance reduction techniques for Monte Carlo methods. We improve standard Monte Carlo methods for computing p-quantiles and multilevel Monte Carlo methods for computing failure probability

Multiscale Methods and Uncertainty Quantification

In this thesis we consider two great challenges in computer simulations of partial differential equations: multiscale data, varying over multiple scales in space and time, and data uncertainty, due to lack of or inexact measurements. We develop a multiscale method based on a coarse scale correction, using localized fine scale computations. We prove that the error in the solution produced by the multiscale method decays independently of the fine scale variation in the data or the computational domain. We consider the following aspects of multiscale methods: continuous and discontinuous underlying numerical methods, adaptivity, convection-diffusion problems, Petrov-Galerkin formulation, and complex geometries. For uncertainty quantification problems we consider the estimation of p-quantiles and failure probability. We use spatial a posteriori error estimates to develop and improve variance reduction techniques for Monte Carlo methods. We improve standard Monte Carlo methods for computing p-quantiles and multilevel Monte Carlo methods for computing failure probability