# 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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57 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: ...
38 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/...
64 views

### Algebraic multigrid as solver and as preconditioner

My question is around the efficiency of AMG. In which case AMG can perform better,as solver or as a preconditioner(for example a Krylov space method as CG)? Assume the case of elliptic pdes.
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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 ...
37 views

### How can the choice of coarsening factor affect Multigrid's convergence?

The linear system $Ax=b$ is coming from the discretization of an elliptic PDE. Multigrid method is used in order to solve it. Suppose $c_0$ is the coarsening factor on level 0 and $c_m$ the coarsening ...
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 ...
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 ...
151 views

### coarsening coefficient matrixes (A2h, A4h...) for geometric multigrid method in 2-D/3-D

I am learning about multigrid methods from the textbook section 6.3 Multigrid Methods, which shows a geometric multigrid algorithm for 1-D examples in detail, including how to build restriction/...
107 views

### Why does smoothed aggregation multigrid method used as preconditioner in conjugate gradient slows down the solution time?

I'm solving a system of linear equations obtained from the FEM discretization of a simple linear elasticity problem on a cube with zero displacements at one plane and a load on the opposite one. The ...
57 views

### Calculating coarse grid matrix in geometric multigird

The coarse grid matrix is calculated via RAP where R,P are the restriction and interpolation matrix,respectively.By checking a typical MG algorithm I want to ask how to calculate efficiently coarse ...
100 views

### Full approximation scheme - smoothers - literature recomendation

I would like to use full approximation scheme(FAS) for solving nonlinear PDE. I am looking into practical implementation of FAS in parallel(MPI) settings. I noticed the most common/effective smoother ...
154 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, ...
81 views

### How the number of pre-smoothing and post-smoothing steps affect the asymtotic convergence rate of geometrical Multigrid?

Does the convergence rate of multigrid depend on the total number of smoothing steps or on the number of pre and post smoothing steps seperately?
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### How coarsening rate affects MG convergence?

If we use an aggresive coarsening how does this affect the convergence of a MG cycle?Is the convergence slower?Do we need more cycles in this case?
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### Why post-smoothing in MG is needed?

In pre-smoothing we eliminate high frequencies,but what about post-smoothing?Is it used for the remain high frequencies that pre-smoothing didn't eliminate?
55 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 ...
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 ...
76 views

### How to implement Geometric Multigrid in non-rectangular grids?

It is quite easy to implement multigrid on a rectangular grid but what about an non-rectangular?How to coarse a non-rectangular grid and apply multigrid(assume an easy non-rectangular grid capital ...
66 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 ...
191 views

### How suitable is multigrid method for time-dependent PDEs?

For elliptic PDEs (Poisson-type), the multigrid method is very sufficient, but how about time-dependent problems (i.e parabolic or hyperbolic PDEs)? Is it efficient to solve such problems using a ...
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 ...
97 views

### Can a direct method like Thomas be used in a multigrid method as a smoother?

As far as I know, multigrid uses stationary iterative methods as smoothers (i.e GS), but can we use a direct method also? For example, in case we have a tridiagonal system (for example 1D heat ...
60 views

### Is there any method to incorporate minor changes into solved meshes to speed convergence in particle-in-cell solvers?

Apologies for the terrible title. I'm trying to perform a 10^6 timestep electrostatic particle-in-cell simulation on a rather large mesh, with very limited computational resources (a single GPU). ...
100 views

### How do multigrid approaches deal with Gibbs phenomenon?

I know (from https://scicomp.stackexchange.com/a/31339/20545, among others) that I need a certain mesh density in FEM, else I might get non-physical oscillations in my solution. How do multigrid ...
165 views

### Null space for smoothed aggregation algebraic multigrid

I do not really get the point of null space usage for creating the prolongation operator for smoothed aggregation algebraic multigrid. I know what the null space is per definition and I know that the ...
131 views

### Algebraic multigrid for complex valued matrices

Assume one uses the classical AMG with Ruge-Stuben coarsening and direct interpolation for solving real valued problems. How can this approach be recycled to also solve complex valued problems like ...
296 views

### Multigrid preconditioner for conjugate gradient methods

When solving $A*x=b$ using preconditioned conjugate gradient methods one has to solve $z=K^{-1}*r$ for the preconditioning where $K$ is the preconditioner of $A$ and $r$ is the residual vector. ...
41 views

### Converting mass density to point mass approximation on a grid

In an nbody gravity simulation, instead of doing exact(all-pair brute force) solution, I added masses of each body into cells of a 3D grid(each cell is just a float value having a mass value). Then ...
515 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 ...
489 views

### Algebraic multigrid in PETSc

Consider a potential/poisson equation on a very large, complicated geometry. Currently, an self-written FEM and linear solvers from NumPy are used. Performance is, of course, not good enough for ...
313 views

### For which problems Krylov subspace methods are preferred over multigrid methods?

As multigrid methods are known to have grid independent convergence rates with $O(N)$ computational cost, then why would one be interested in using Krylov subspace methods at all, for which ...
311 views

### V-cycle Multigrid for 2D transient heat transfer on a square plate using finite difference

I'm currently developing a program to solve 2D transient state heat conduction on a square plate using the V-cycle multigrid. Althought my program is able to reach the steady state solution, it's ...
324 views

### What is the default smoother for the "PCMG" preconditioner in PETSc?

For a large parallel sparse matrix (mpiaij type matrix) in my code, I was experimenting with various preconditioners to see which one would do best with GMRES/BiCGSTAB. I tried the PCMG ...
133 views

### Can I use multigrid to solve linear algebra problems that do not arise from a differential equation?

Can the Multigrid method be applied to solve a linear system of equations in the form $Ax=b$, that is not necessarily related to differential equations?
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### Is it usual to have no convergence checking in Multigrid?

I just read Chapter 3 in "A Multigrid Tutorial" by Briggs/Henson/McCormick, link. The text is about Multigrid cycles such as V-cycle, mu-cycle, FMG. What caught my eye: In most iterative procedures ...