A Localized Orthogonal Decomposition Method for Semi-linear Elliptic Problems

Galerkin Proper Orthogonal Decomposition Methods for Parameter Dependent Elliptic Systems

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of linear and non-linear systems. It is based on a Galerkin type discretization with basis elements created from the system itself. In this work, error estimates for Galerkin POD methods for linear elliptic, parameter-dependent systems are proved. The resulting error bounds depend on the number of POD basis functi...

Matrix Decomposition Algorithms in Orthogonal Spline Collocation for Separable Elliptic Boundary Value Problems

Fast direct methods are presented for the solution of linear systems arising in highorder, tensor-product orthogonal spline collocation applied to separable, second order, linear, elliptic partial di erential equations on rectangles. The methods, which are based on a matrix decomposition approach, involve the solution of a generalized eigenvalue problem corresponding to the orthogonal spline co...

A reduced basis localized orthogonal decomposition

In this work we combine the framework of the Reduced Basis method (RB) with the framework of the Localized Orthogonal Decomposition (LOD) in order to solve parametrized elliptic multiscale problems. The idea of the LOD is to split a high dimensional Finite Element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information. ...

Shape-Measure Method for Solving Elliptic Optimal Shape Problems (Fixed Control Case)

A rigorous Multiscale Method for semi-linear elliptic problems

In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations in small patches that have a diameter of o...