Related papers: Multigrid Primer: Basic Principles
In recent publications, the author and his coworkers have shown robust approximation error estimates for B-splines of maximum smoothness and have proposed multigrid methods based on them. These methods allow to solve the linear system…
We propose V--cycle multigrid methods for vector field problems arising from the lowest order hexahedral N\'{e}d\'{e}lec finite element. Since the conventional scalar smoothing techniques do not work well for the problems, a new type of…
We discuss how multi-grid computing schemes can be used to design hierarchical coordination architectures for energy systems. These hierarchical architectures can be used to manage multiple temporal and spatial scales and mitigate…
The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the structure and Eulerian variables to describe the fluid. Explicit time stepping schemes for the IB method require…
We design and analyze multigrid methods for the saddle point problems resulting from Raviart-Thomas-N\'ed\'elec mixed finite element methods (of order at least 1) for the Darcy system in porous media flow. Uniform convergence of the…
We propose a multi-level method to increase the accuracy of machine learning algorithms for approximating observables in scientific computing, particularly those that arise in systems modeled by differential equations. The algorithm relies…
The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the…
We develop a simple algorithmic framework to solve large-scale symmetric positive definite linear systems. At its core, the framework relies on two components: (1) a norm-convergent iterative method (i.e. smoother) and (2) a preconditioner.…
Unfitted finite element methods have emerged as a popular alternative to classical finite element methods for the solution of partial differential equations and allow modeling arbitrary geometries without the need for a boundary-conforming…
The principle of indirect elimination states that an algorithm for solving discretized differential equations can be used to identify its own bad-converging modes. When the number of bad-converging modes of the algorithm is not too large,…
The convergence of multigrid methods degrades significantly if a small number of low quality cells are present in a finite element mesh, and this can be a barrier to the efficient and robust application of multigrid on complicated geometric…
We introduce a novel technique for proving global strong discrete maximum principles for finite element discretizations of linear and semilinear elliptic equations for cases when the common, matrix-based sufficient conditions are not…
In this paper, we present a multigrid preconditioner for solving the linear system arising from the piecewise linear nonconforming Crouzeix-Raviart discretization of second order elliptic problems with jump coefficients. The preconditioner…
We explore a reinforcement learning strategy to automate and accelerate h/p-multigrid methods in high-order solvers. Multigrid methods are very efficient but require fine-tuning of numerical parameters, such as the number of smoothing…
In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for…
One of the most important goals of the 21st century is to change radically the way our society produces and distributes energy. This broad objective embodies in the smart grid's futuristic vision of a completely decentralized system powered…
This paper deals with distributed control of microgrids composed of storages, generators, renewable energy sources, critical and controllable loads. We consider a stochastic formulation of the optimal control problem associated to the…
In electrical distribution grids, the constantly increasing number of power generation devices based on renewables demands a transition from a centralized to a distributed generation paradigm. In fact, power injection from Distributed…
In this work, we propose a multigrid preconditioner for Jacobian-free Newton-Krylov (JFNK) methods. Our multigrid method does not require knowledge of the Jacobian at any level of the multigrid hierarchy. As it is common in standard…
When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…