Daniel Elfverson

Convergence of a Discontinuous Galerkin Multiscale Method

We present a discontinuous Galerkin multiscale method for second order elliptic problems and prove convergence. We consider a heterogeneous and highly varying diffusion coefficient in

An Adaptive Discontinuous Galerkin Multiscale Method for Elliptic Problems

L^\infty(\Omega,\mathbb{R}^{d\times d}_{sym})

On multiscale methods in Petrov---Galerkin formulation

with uniform spectral bounds 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 the 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

A multilevel Monte Carlo method for computing failure probabilities

H

Shape optimization using the cut finite element method

to a reference solution, it is sufficient to choose the patch sizes

Uncertainty quantification for approximate p-quantiles for physical models with stochastic inputs

\mathcal{O}(H|\log H|)

A cut finite element method for the Bernoulli free boundary value problem

. We also discuss a way to further localize the corrected basis to elementwise support. Improved convergence rate can be achieved depending on the piecewise regularity of the forcing function. Linear convergence in energy norm and quadratic convergence in the

CutIGA with basis function removal

L^2

A discontinuous Galerkin multiscale method for convection–diffusion problems

-norm is obtained independently of the ...

Cut Topology Optimization for Linear Elasticity with Coupling to Parametric Nondesign Domain Regions

An adaptive discontinuous Galerkin multiscale method driven by an energy norm a posteriori error bound is proposed. The method is based on splitting the problem into a coarse and fine scale. Localized fine scale constituent problems are solved on patches of the domain and are used to obtain a modified coarse scale equation. The coarse scale equation has considerably less degrees of freedom than the original problem. The a posteriori error bound is used within an adaptive algorithm to tune the critical parameters, i.e., the refinement level and the size of the different patches on which the fine scale constituent problems are solved. The fine scale computations are completely parallelizable, since no communication between different processors is required for solving the constituent fine scale problems. The convergence of the method, the performance of the adaptive strategy, and the computational effort involved are investigated through a series of numerical experiments.