# 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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115 views

### Infinite Function Value on Dirichlet Boundary

I have been working on a multigrid solution to a non-homogeneous Dirichlet boundary value problem. However, the function goes to infinity on the boundary. This causes numerical overflow errors to be ...
55 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 ...
464 views

### How to define residual in multigrid approach?

I wish to solve the two-dimensional Navier Stokes equations using multigrid method on a collocated grid using the Predictor-Corrector method mentioned below. But first, let me elaborate on what I had ...
149 views

### Solve FEM matrix from coupled system

I'm developing an FEM solver for a coupled system. I have diffusion and potential equations which result in positive definite matrices for each equation, but the coupling makes the overall system ...
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 ...
136 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, ...
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 ...
62 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 ...
271 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. ...
56 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.
58 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). ...
97 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 ...
77 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?
55 views

### 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?
187 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 ...
458 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 ...
244 views

### FENICS subdomains - restriction/ prolongation operators

I am trying to implement my own multigrid method in fenics. Is there any "smart/ fenics" way how to assemble subdomains and obtain restriction/ prolongation operators ? Thanks!
173 views

### Specific questions for 2-D Multigrid

I am simulating $\nabla^{2}u=0$ with mixed Dirichlet-Neumann boundary conditions on 2-D using 2-Grid method. Dirichlet ...
304 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 ...
123 views

### A doubt in Multigrid V-cycle

Assume I have 3 levels of grids. Finest Grid = level 2, Coarser Grid = level 1, Coarsest Grid = level 0. Relax $u$ on $Au = b$ at level 2 for 3 times. Find residual $r2$ at level 2, then restrict to ...
34 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/...
51 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 ...
33 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 ...
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 ...
308 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 ...
84 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 ...
215 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 ...
50 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 ...
204 views

### What libraries provide an implementation of multigrid?

I am working on numerical method of Multigrid. What's the available implementation(solver) (actually used in scientific computation) of multigrid method?
39 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 ...
969 views

I would like to code for a quadtree type meshing but don't know how to do. If anyone can help or can share any starting code?
114 views

### How to determine the number of c points in algebraic multi grid

I am trying to write an algebraic multi-grid solver (in c++). At a given level I determine which nodes are c-points and which nodes are f-points (where the total number of c and f points equals the ...
74 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 ...
48 views

### Clarification on interpolation equalities given by Briggs

Briggs, "A Multigrid Tutorial" (pg. 35) has the following expressed as 2-D interpolation: \begin{align*} v^h_{2i,2j} &= v_{i,j}^{2h}\\ v^h_{2i+1,2j} &= 0.5\cdot(v_{i,j}^{2h} + v_{i+1,j}^{2h})\\...
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 ...