# Multilevel decompositions and norms for negative order Sobolev spaces

@inproceedings{Fuhrer2020MultilevelDA, title={Multilevel decompositions and norms for negative order Sobolev spaces}, author={Thomas Fuhrer}, year={2020} }

We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in H−s for s ∈ (0, 1). Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the…

## 6 Citations

### On a mixed FEM and a FOSLS with $H^{-1}$ loads

- Mathematics
- 2022

. We study variants of the mixed ﬁnite element method (mixed FEM) and the ﬁrst-order system least-squares ﬁnite element (FOSLS) for the Poisson problem where we replace the load by a suitable…

### HAZniCS - Software Components for Multiphysics Problems

- Computer ScienceArXiv
- 2022

The focus of the paper is on the design and implementation of a pool of robust and efficient solver algorithms which tackle issues related to the complex interfacial coupling of the physical problems often encountered in applications in brain biomechanics.

### Least squares solvers for ill-posed PDEs that are conditionally stable

- MathematicsArXiv
- 2022

A BSTRACT . This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least…

### MINRES for second-order PDEs with singular data

- MathematicsSIAM J. Numer. Anal.
- 2022

This work considers a DPG method and a least-squares FEM for the Poisson problem and analyzes regularization approaches that allow the use of H−1 loads, and also studies the case of point loads.

### Robust monolithic solvers for the Stokes-Darcy problem with the Darcy equation in primal form

- Computer ScienceSIAM Journal on Scientific Computing
- 2022

This work constructs mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem and suggests robust preconditioners that utilize operators in fractional Sobolev spaces.

### Interpolation Operator on negative Sobolev Spaces

- MathematicsArXiv
- 2021

A Scott–Zhang type projection operator mapping to Lagrange elements for arbitrary polynomial order that is stable in the corresponding negative norms and allows for optimal rates of convergence.

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