Axel Målqvist

Localization of elliptic multiscale problems

This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size

A posteriori error estimates for mixed finite element approximations of parabolic problems

H

Multiscale methods for elliptic problems

, patches of diameter

Computation of eigenvalues by numerical upscaling

H\log (1/H)

Convergence of a Discontinuous Galerkin Multiscale Method

are sufficient to preserve a linear rate of convergence in

Localized Orthogonal Decomposition Techniques for Boundary Value Problems

H

A localized orthogonal decomposition method for semi-linear elliptic problems

without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods. - See more at: http://www.ams.org/journals/mcom/2014-83-290/S0025-5718-2014-02868-8/#sthash.z2CCFXIg.dpuf

Two-Level Discretization Techniques For Ground State Computations Of Bose-Einstein Condensates

We derive residual based a posteriori error estimates for parabolic problems on mixed form solved using Raviart–Thomas–Nedelec finite elements in space and backward Euler in time. The error norm considered is the flux part of the energy, i.e. weighted L2(Ω) norm integrated over time. In order to get an optimal order bound, an elementwise computable post-processed approximation of the scalar variable needs to be used. This is a common technique used for elliptic problems. The final bound consists of terms, capturing the spatial discretization error and the time discretization error and can be used to drive an adaptive algorithm.

Adaptive variational multiscale methods based on a posteriori error estimation : Duality techniques for elliptic problems

In this paper we derive a framework for multiscale approximation of elliptic problems on standard and mixed form. The method presented is based on a splitting into coarse and fine scales together with a systematic technique for approximation of the fine scale part, based on the solution of decoupled localized subgrid problems. The fine scale approximation is then used to modify the coarse scale equations. A key feature of the method is that symmetry in the bilinear form is preserved in the discrete system. Other key features are a posteriori error bounds and adaptive algorithms based on these bounds. The adaptive algorithms are used for automatic tuning of the method parameters. In the last part of the paper we present numerical examples where we apply the framework to a problem in oil reservoir simulation.

Nonparametric Density Estimation for Randomly Perturbed Elliptic Problems I: Computational Methods, A Posteriori Analysis, and Adaptive Error Control

We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we consider benchmark multi-scale eigenvalue problems in reservoir modeling and material science. We compute a low-dimensional generalized (possibly mesh free) finite element space that preserves the lowermost eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of