Questions tagged [multigrid]

An approach to solving systems of equations by projecting the problem from a fine scale representation onto a coarser one. A coarse representation generally has fewer unknowns, making it faster to solve than the original problem. The coarse solution can then be projected back onto the finer problem as an initial guess of the solution to the finer problem.

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Construction of Prolongation and Restriction Operator for Geometric Multigrid (2D-FEM): Resulting in a Decreasing Solution

Consider the following problem, $$ -\Delta u(x) = f(x), \qquad x \in \Omega \\ u(x) = 0,\qquad x \in \partial \Omega$$ with $\Omega = [0,1]\times [0,1]$ being the domain and $\partial \Omega$ being ...
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165 views

Geometric Multigrid for Conform and Non–″ Elements: Restriction Operators

First of all, let me set up some notations. Suppose we have a hierarchy of meshes $\mathcal{T}_0 \subset \mathcal{T}_1 \subset \dots \subset \mathcal{T}_n$. For simplicity I restrict myself here to $...
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535 views

Neumann-Neumann boundary intersection

I am trying to solve a Vertex Centered, Finite Difference, Poisson equation $-\nabla^{2}u=f$ with Dirichlet boundaries on the left and bottom and Neumann boundaries on the top and right using ...
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196 views

Multigrid for Robin boundary conditions

I have a question regarding the treatment of Robin boundary conditions in a multigrid solver. I am solving the Poisson equation in $\Omega=(0,1)^2$ with Robin boundary conditions on the boundary, $$- \...
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1answer
231 views

Can F-cycle substitue FMG for update of existent solution?

I have a nicely working multigrid solver, which I use for solving the Poisson equation from an electrostatic problem. I solve this equation first without any charges, and then many times with a slowly ...
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66 views

Multigrid method: linear solver and modified residual

I am trying to better understand the FAS multigrid algorithm for Euler equation in FV discretization. The usage of the modified residual (the residual with forcing) inside the different cases: ...
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151 views

Parallel In Time with Multigrid

I am trying to solve the linear finite element equation $M\ddot{u}+Ku=F(t)$, where $M$ is the mass matrix ,$K$ the stiffness matrix and $F(t)$ the external load vector, parallel in time using XBraid ...
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172 views

Residual of Poisson equation with periodic boundaries

I am trying to write a multigrid solver for Poisson's equation, $-\Delta u=f$, on the unit square, $\Omega=(0,1)^2$ with periodic boundaries. My primary source has been Multigrid by Trottenberg, ...
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56 views

Avoid matrix multiplication in algebraic multigrid method

Currently when I try to solve a linear algebra system of the form of $A x =b$ I use the algebraic multigrid method. The algebraic multigrid method uses a Galerkin product to form the coarse grid ...
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67 views

Multilevel minimization - boundary conditions

I am interested in minimizing $$min_{x \in R^{n^l}} f^l(x),$$ where $f^l(x)$ is nonlinear objective function arising from discretization of PDE. I would like to use nonlinear multilevel minimization ...
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39 views

Prolongation and restriction operators in multigrid for high order PDEs

If I have the Poisson equation $\Delta u = f$ a standard transfer operator (for a regular grid) is the full weighting/bilinear interpolation scheme: $$K = \frac{1}{4}\begin{bmatrix}\frac{1}{4} & \...
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40 views

Multigrid: Gauss–Seidel eigenvalues and eigenvectors

I am trying to figure out the questions from Multigrid Tutorial by Briggs. However I am stuck on these two questions. https://www.researchgate.net/publication/...
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52 views

Multigrid Reduction In Time Convergence

I am trying to solve a 2D dynamic linear elasticity model parallel in time using Xbraid. The spatial domain is [0,1]x[0,1] and time domain [0,1]. For time integration I am using a backward Euler ...
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34 views

What kind of problem or matrices are suitable for multigrid method?

For Poisson or Convection-diffusion equation as follows: $$ -\Delta u=f,\qquad u|_\Omega = g. $$ or $$ -\Delta u +\vec{w}.\nabla u=f,\qquad u|_\Omega = g. $$ using FDM or FEM discretization, we can ...
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60 views

Can the standard multigrid performance be used for time-dependent PDEs?

Consider a time dependent pde(i.e u(x,t)).I know when only space-coarsening is used the standard multigrid performance can be applied but what if instead we use only time-coarsening?Can we apply the ...
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85 views

What does it take to prove that a multigrid algorithm scales linearly with system size?

It is my understanding that the multigrid solution techniques are generally the preferable method to solve large Poisson problems. Now assume I have written a multigrid solver that is tailored to my ...
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233 views

Algorithm for Adaptive Mesh Refinement

I am trying to implement Adaptive Mesh Refinement. I am not a Mathematics/Computational Science person so I will try to write the algorithm in a simpler way. I will be grateful if experts can comment ...
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51 views

Optimization of nonlocal stencil-like operator on subset of regular grid

I am trying to optimize the execution time for this particular piece of fortran code. Details: i_gc is a (ngpts, 3) array of containing (i,j,k) indices for each grid point. This is a subset of the ...
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27 views

Restriction in (geometric) multigrid for vectors of non-even length

Naive restriction operators in geometric multigrid that I have seen are typically implemented as a convolution and a subsequent averaging of every two entries in a vector $v^h$. For example: $$\tilde{...
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32 views

Differences between Two-Grid Correction and V-Cycle Scheme

I found in A Multigrid Tutorial: Second Edition by William L. Briggs‏, Van Emden Henson‏, Steve F. (here) the following Schemes: Two-Grid Correction Scheme, p. 37: And V-Cycle Scheme, p. 40 I want ...