Matlab multigrid solver We will show that this can also be achieved for some reaction-diffusion equations on uniformly Nov 20, 2022 · 代数多重网格（Algebraic Multigrid, AMG）是一种高效的数值求解线性系统的预处理技术，尤其适用于大规模的、不规则的稀疏矩阵问题。AMG方法起源于几何多重网格（Geometric Multigrid, GMG），但与GMG不同的是，AMG is that multigrid can solve many sparse and realistic systems to high accuracy in a fixed number of iterations, not growing with n. In order to solve this equation we apply Gauss-Seidel a few times on a full size grid, before restricting the problem to a smaller grid and repeating the process. To solve the Stokes system, the Matlab codes on the single. Choose the dimension you are aiming for, and start from: TestCycle. CMG is originally implemented in MATLAB. Multigrid preconditioned CG for the Poisson equation on rectangular grids can be found in [Tat93] and the algorithm is parallelized in [TO94] and later [AF96]. pdf. Briggs Presented by Solve the ith equation for holding other variables fixed: −ui − +ui −ui + =h f i ≤i ≤N − u0 =uN = 2 Dec 1, 2011 · In this work we describe the Combinatorial Multigrid (CMG) solver that provides an affirmative answer. A test case can be found at AMG_test. This MATLAB implementation can provide a valuable resource for researchers and practitioners applying the multigrid method to solve the diffusion equation. For a more detailed mathematical introduction to the multigrid algorithm, see "A Multigrid Tutorial" by W. Feb 5, 2023 · Implementation of the Multigrid Method (MG) for solving Ax = b, uses Gauss-Seidel or Jacobi for smoothing. classical iterative methods 2. Reference: Jul 28, 2022 · 文章浏览阅读1. algebraic multigrid linear solver This program solves Ax=b where A is an M-matrix. It is hosted here. For 2D version, we used a mixture of Jacobi and Gauss-Seidel iteration with conservative finite difference as the smoother. Examples of solvers that are matrix-free: All Krylov-subspace solvers are matrix-free. The library is well suited for implicit unstructured methods. Oct 30, 2019 · I have heard that in matlab, there is a function 'fft' (fast fourier transformation) can solve Ax=b quickly, but I can not understand that. (2) Not optimal with standard multigrid V- or W-cycles (the number of iterations grows with the number of levels and hence with the matrix size) Solution: Enhanced multigrid cycle: the K-cycle In a multigrid algorithm, the coarse systems Acuc =rc are approximately solved with a few iterations of the two-grid method at the considered (coarse) level In order to implement a solver for these equations, multigrid method is introduced to our solve. arising from the discretization of second order elliptic PDEs. figures: Figures used in the report; ma_solver. multigrid_poisson_1d_test. A Julia implementation has been co-authored by Bodhisatta Pramanik. This solver type is based on a discrete sensitivity method, which provides enhanced robustness, improved accuracy, and faster performance for gradient-based optimization with the Time-Dependent Solver. • There are 2 basic multigrid approaches − geometric and algebraic • In geometric multigrid, the geometry of the problem is used to define the various multigrid components. Multigrid preconditioning is used in practice even for linear systems, typically 如此循环往复。因为先由细到粗，后由粗到细，网格转换的路径形似 V 字，所以该方法被称为 V-Cycle Multigrid。除此之外还有 F-cycle multigrid 和 W-cycle multigrid 。他们的基本思想都是相同。 4. MATAMG support classical algebraic multigrid(AMG) interpolation, adaptive AMG(aAMG) interpolation and bootstrap AMG(BAMG) interpolation. This code is completely in OOP. We begin by stating the algorithm at a high level, and then fill in details. This reduces to a matrix equation . Can someone give me the matlab impelementation codes that I can easily understand how matlab use 'fft' to solve this Ax=b quickly? Thanks very much. Download, take a look at readme, and install. The skeleton of the code is the same as the perfect 2D multigrid solver provided by Achi Brandt. Porous convection can describe migration of ground water and hydrocarbons in the earth‟s crust. algebraic multigrid linear solver In this paper, we try to implement a GPU solver for Stokes Equations with variable viscosity based on CUDA using geometric multigrid methods on the staggered grids. The direct solver methods implemented in mldivide can be used to solve distributed systems of linear equations in parallel but may not be efficient for certain large and sparse systems. Furthermore, Amir et al. The method uses two grid recursively using Gauss-Seidel for smoothing and elimination to solve at coarsest level. Jan 15, 2025 · Moreover, our proposed MATLAB implementation of the multigrid method presented in this paper is concise, consisting of ∼90 lines of code. The Multigrid_Solver() will first call Multigrid{1,2,3}D_Vcycle_GenMat() to generate the coefficient matrices and restriction operators on each level and store them, then it will call Multigrid_Vcycle() to perform V-cycle computation until the relative residual norm is smaller than the given threshold. Files The following files are included in this repository: V_cycle. List the iteration steps and CPU time for different size of matrices. It is perfect for students because it was written by a graduate student. 14 Multigrid implementation using Matlab Simple, pedagogical Matlab implementation of the Multigrid method for solving Poisson-like equations. - xinwangmath/multigrid In the following, we are describing the geometric multigrid method, which for certain problems yields an iterative solver with optimal cost complexity, i. Reference: Jan 1, 2020 · Liu and Tovar [53] presented an efficient 3D TO code using the PCG in Matlab. I am using the following command to get the solution: x=A\\B; This command find the va Feb 26, 2024 · 多重网格技术（multigrid solver） 微分方程的误差分量可以分为两大类，一类是频率变化较缓慢的低频分量；另一类是频率高，摆动快的高频分量。一般的迭代方法可以迅速地将摆动误差衰减，但对那些低频分量，迭代法的效果不是很显著。 The most basic form of multigrid inolves solving the Poisson equation by finite differences on a grid. In this context, multigrid meth-ods [39] are among the most efficient solution techniques. The main focus in this course will be on multigrid methods suitable for finite difference, finite Multigrid Methods#. Finally, the computing times for the application of the sparse direct solver UMFPACK, Davis (2004), are given. conjugate gradient (CG), LGMRES, BICG, etc. in matlab Matrix-free solvers are thus a subclass of available solvers. Apr 9, 2021 · Download and share free MATLAB code, including functions, models, apps, support packages and toolboxes A Geometric Multigrid Solver for Large-scale Static Finite In this paper we present a software package which implements an efficient multigrid sparse solver running on Graphics Processing Units. In my case, I am making simple multigrid i. Multigrid solver for matlab. To acheive multigrid efficiency, a hierarchical ‘grids’ is generated from the graph of A. run; Jul 7, 2014 · INTRODUCTION ===== MATAMG stands for MATLAB Algebraic Multigrid. Olson and Jacob Schroder and Ben Southworth}, title = {{PyAMG}: Algebraic Multigrid Solvers in Python}, journal = {Journal of Open Source Software}, year = {2023}, publisher = {The Open Journal}, volume = {8}, number = {87}, pages = {5495}, doi = {10. , Aage, N. The library includes a flexible solver composition system that allows a user to easily construct complex nested solvers and preconditioners. A new Time discrete adjoint solver type is available for optimal control and time-dependent parameter estimation. Another achievement in the formulation of multigrid methods was the full multigrid (FMG) scheme [4, 21], based on the combination of nested iteration techniques and multigrid methods. Multigrid algorithms are now applied to a wide range of The overall goals of this project are to parallelize an existing serial code (C/C++) for a multigrid poisson equation solver using MPI and to study the performance and scalability of the resulting implementation. (2014). [KH08] introduced a higher-order parallel multigrid solver for large rectangular images. The solver is based on a Finite Volume (FV) discretization of the Reynolds equation incorporating mass-conserving cavitation through the cavity fraction and elastic deformation through the application of the Boundary Element Method (BEM) to an elastic half-space. This will require the parallelization of two key components in the solver: 1. This work has been supported by NSF grant CCF-#1149048 Sep 10, 2013 · Download and share free MATLAB code, including functions, models, apps, support packages and toolboxes is that multigrid can solve many sparse and realistic systems to high accuracy in a xed number of iterations, not growing with n. edu/~seibold seibold@math. Usage instructions are included in the README. Apr 9, 2021 · Download and share free MATLAB code, including functions, models, apps, support packages and toolboxes A Geometric Multigrid Solver for Large-scale Static Finite Sep 17, 2022 · Afivo用到了multigrid方法求解PDE方程，需要先简单理解一下multigrid的原理，找到了视频教程和对应的octave代码，先在博客中记录备忘。可见multigrid（MG）方法在求解大型线性方程组具有一定的优越性。 AGMG solves systems of linear equations with an aggregation-based algebraic multigrid method. Start with the to understand how multigrid works. m Benjamin Seibold Applied Mathematics Massachusetts Institute of Technology www-math. It is suitable for small size problems. S. Combinatorial Multigrid is a solver for symmetric diagonally dominant linear systems. contour function to show temperature on a 2d plate at 0, 1, 2, 5, 10, 100 seconds for discretised pde with jacobi solver. E. The purpose of this repository is to provide Matlab code for geometric multigrid that is easy to understand and learn from. Feb 1, 2014 · Implementation of a multigrid solver on a GPU for Stokes equations with strongly variable viscosity based on Matlab and CUDA. In this thesis it is shown that it can be used in an application where porous convection is simulated, see Figure 1. where u is the solution, f is a given function, and ∇^2 is the Laplace operator. In order to implement a solver for these equations, multigrid method is introduced to our solve. CMG: Combinatorial Multigrid. This package lets you solve sparse linear systems using Algebraic Multigrid (AMG). Code is run from main. Discrete Adjoint Solver Type. For a matlab implementation of multigrid click here. What differentiates CMG from other multigrid solvers is its setup phase which is based on a sound algebraic machinery. A multigrid solver for 2D Poisson equation, implemented in Matlab. We briefly describe one such approach, Newton-multigrid, in the following section. The number of pre- and postsmoothing and coarse grid iteration steps can be prescribed. ° Can solve such a coarse approximation to get an approximate solution, iterating if necessary • Solve coarse approximation problem by using an even coarser approximation of it, and so on recursively ° Ex: Multigrid for solving PDE in O(n) time • Use coarser mesh to get approximate solution of Poisson’s Eq. e. The solver is written in MATLAB but it will be soon offered in other environments. Jun 15, 2022 · MATLAB solver for Elastohydrodynamic Lubrication (EHL) problems. Solve a square linear system using pcg with default settings, and then adjust the tolerance and number of iterations used in the solution process. 以最简单的 V-Cycle 为例 code_matlab: Folder contains the MATLAB version of the code used. One is to use multigrid as the linear solver in a standard linearization, such as in Newton’s method or Picard iteration. 1. 21105/joss of iterations with a multigrid method as solver and with a multigrid method as preconditioner within a exible general minimized residual (GMRES) method are presented. 5 AGMG solves systems of linear equations with an aggregation-based algebraic multigrid method. A restriction matrix R transfers vectors from the ne grid to the coarse Sep 17, 2023 · Download and share free MATLAB code, including functions, models, apps, support packages and toolboxes Block tridiagonal method was used to solve the problem on Nov 22, 2017 · PROGRAMMING OF MULTIGRID METHODS LONG CHEN In this note, we explain the implementation detail of multigrid methods. I know that A is a sparse matrix. using pointers. m: Implements the V A multigrid method with an intentionally reduced tolerance can be used as an efficient preconditioner for an external iterative solver, e. We will use the approach by space decomposition and subspace correction method; see Chapter: Subspace Correction Method and Auxiliary Space Method. Sep 5, 2013 · Hi everybody, I have to solve a linear system of equations (Ax=B). Follow the Step 3 in part 2 to code a V-cycle. This works especially well for symmetric positive definite matrices. Fixed point iterations. 5k次。Afivo用到了multigrid方法求解PDE方程，需要先简单理解一下multigrid的原理，找到了视频教程和对应的octave代码，先在博客中记录备忘。可见multigrid（MG）方法在求解大型线性方程组具有一定的优越性。. It is expected to be efficient for large systems arising from the discretization of scalar second order elliptic PDEs. Find and download the corresponding . Multigrid method is commonly used in reducing the iteration steps for solving the elliptic partial differential equation with the ill-conditioned matrix due to the saddle points in the matrix system coupling mass and momentum equations and strongly Multigrid solver for 1d Poisson problem: mit18086_multigrid. lyze our solver’s efficiency on a GPU. Feb 24, 2013 · matlab经典小代码经典代数多重网格方法（AMG）演示 经典代数多重网格方法（AMG）的简单实现。多重网格求解器的主要过程和求解器中的参数（如平滑器前后的数量）与几乎相同。 A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. Chebyshev iteration. com Jun 8, 2021 · 1D/2D/3D finite difference multigrid solver on a regular Cartesian grid. All solvers were called from the Matlab environment Aug 10, 2011 · Multigrid Solver for scalar elliptic linear PDEs The PDE specifications need to have the format used by the PDE toolbox. Nov 14, 2013 · This article presents a computational approach that facilitates the efficient solution of 3-D structural topology optimization problems on a standard PC. geometric multigrid Oct 13, 2014 · 多重网格技术（multigrid solver） 微分方程的误差分量可以分为两大类，一类是频率变化较缓慢的低频分量；另一类是频率高，摆动快的高频分量。一般的迭代方法可以迅速地将摆动误差衰减，但对那些低频分量，迭代法的效果不是很显著。 There are two basic approaches to using multigrid in the solution of (1. [54] combined multigrid method and PCG into multigrid preconditioned conjugate gradients (MGCG), which can reduce the solving time of the linear equations for large-scale TOs. Geometric and Algebraic Multigrid 18 • One of the most important issues in multigrid is the construction of the coarse grids. g. The LOR solvers miniapp provides matrix-free solvers for the same problems solved in Examples 1, 3, and 4. The solver can be used to solve the Poisson equation of the form: ∇^2 u = f. The matrix formulation will be obtained This code provides a MATLAB implementation of a 2D Poisson solver using the multigrid method. Then use the V-cycle as a preconditioner in PCG. m. It is MATLAB toolbox designed to solve a linear system with algebraic multigrid algorithms. • However, we would like to us multigrid ideas to treat the nonlinearity directly. , & Lazarov, B. Test the robustness of the solver, apply uniformrefine to a mesh and generate corresponding matrix. Briggs, SIAM, 1987. (1) Given a uniform grid T_N of (0,1), we shall test several iterative methods for solving the linear system obtained by linear finite element approximation of (1). Sep 10, 2013 · Download and share free MATLAB code, including functions, models, apps, support packages and toolboxes. For linear equations in graph Laplacians, Livne and Brandt presented a variant of AMG solver, called the Lean Algebraic Multigrid (LAMG), and provided a MATLAB imple-mentation [LB12]. Source Code The reviewed source code and documentation of a Matlab implementation for Multigrid Poisson solvers and the applications described in this work are available fromthe web page of this article1. _多重网格法matlab CG rather rather than a stand-alone solver can alleviate in-stability issues on some problems. This repository includes matlab codes that were used in the following papers: Amir, O. For the large-scale ITO, solving equations costs a large part of Sep 10, 2013 · Download and share free MATLAB code, including functions, models, apps, support packages and toolboxes Multigrid Tutorial By William L. To get started, run convergence_plot() in Matlab. . I have solved similar equations in COMSOL with iterative solvers like Conjugate Gradient and Algebraic Multigrid Preconditioners. The post-processing plot has the name dummy. the solver returns a solution to a PDE in \(O(n_{\text{DoFs}})\) arithmetic operations. Go to the LOR solvers miniapp directory: cd ~/mfem/miniapps How does multigrid fit in? • One obvious method is to use multigrid to solve J(v)e = r at each iteration step. When the mesh is avaiable, x = mg(A,b,elem) implements geometric multigrid solvers Example 26 showed how to use geometric multigrid together with matrix-free methods. When the mesh is avaiable, x = mg(A,b,elem) implements geometric multigrid Feb 5, 2018 · I coded multigrid solver for Poisson equation in matlab. 2h-Solve(also a v-cycle): 2h-Iterate Smoothing; 2h\rightarrow 4h-Restrict; 4h Solve; 4h \rightarrow 2h-Interpolate; 2h-Iterate Smoothing; 2h\rightarrow h-Interpolate; h-Iterate; 我们在分析的时候，为了方便起见，认为误差并没有经过 2h-Iterate的两步，否则整个分析过程会变的相当复杂。 Multigrid methods are very efficient iterative solvers for large systems of linear and nonlinear algebraic equations. The geometry description 'g' and the boundary condition description 'b' can either be the name of a function file (see the Matlab help to pdegeom and pdebound for g and b, respectively) or matrices (see decsg and assemb, respectively). The cost reduction is obtained by exploiting specific characteristics of a multigrid preconditioned We consider the Poisson equation in 1-D: -u'' = f in (0,1) and u(0)=u(1)=0. 3. MGMRES, a MATLAB library which applies the restarted GMRES algorithm to solve a sparse linear system, by Lili Ju. txt le of the archive. 11 years ago | 9 downloads | iFEM is a MATLAB software package containing robust, efficient, and easy-following codes for the main building blocks of adaptive finite element methods on unstructured simplicial grids in both two and three dimensions. Step 4 V-cycle Multigrid used with PCG. Keywords GPU, Matlab, multigrid, Stokes flow, strongly variable viscosity 1 Introduction Graphics processing units (GPUs) are increasingly being used to solve numerical problems, since NVIDIA first released the Compute Unified Device Architecture (CUDA) in 2007. • Hence, we need to specialize the multigrid components (relaxation, grid transfers, Submitted. The driver provided here builds linear systems for various 3-dimensional problems. Everyone who is new to this should learn how masters do their work. 1D/2D/3D finite difference multigrid solver on regular grid. CMG combines the strengths of multigrid with those of combinatorial preconditioning. Besides the simplicity and readability, sparse matrixlization, an innovative programming style for MATLAB, is introduced to improve the efficiency. edu March 31, 2008 1 Introduction On the following pages you find a documentation for the Matlab AMG solver applicable to Dirac equations [Bra00]. coarse grid correction cycle using 2 levels (fine and coarse grid). mit. AmgX is a GPU accelerated core solver library that speeds up computationally intense linear solver portion of simulations. Feb 13, 2013 · Solving the residual equation on the coarse grid is not at all different from solving a new equation. 'n' , the relaxations needed to get the required accuracy also increases. Optimal multigrid methods can solve linear systems in O(N) number of operations, with N the number of unknowns. mgmres, a MATLAB code which applies the restarted GMRES algorithm to solve a sparse linear system, by Lili Ju. To acheive multigrid efficiency, a hierarchical 'grids' is generated from the graph of A. 1). Direct solvers; need to access \(M_{ij}\) directly Backslash \ is the build-in direct solver of MATLAB. Another multi-grid variant specialized for graph Laplacians was developed by Napov and Notay [NN17]. Computing time associated with solving the nested analysis problem is reduced significantly in comparison to other existing approaches. In this novel coding style "GMGS-3D" proceeds the static Finite Element Analysis (FEA) for solid objects discretized into Cartesian mesh, where, 1) an element index based data structure is used to store the FEA stiffness matrix; 2) combined with the Jacobian smoother, a Geometric Multigrid based V-cycle is built on the Cartesian mesh; 3) the FEA equation is iteratively solved by Conjugate Gradient Method preconditioned Background removal for MRI phase data by solving the Laplacian boundary value problem (full multigrid solver, C++ implementation with Matlab wrapper) - nosarthur/LBV MATLAB codes for efficient 3-D topology optimization Codes use multigrid CG, approximate sensitivity analysis, recycling precoditioners. , [7] The solution may still be obtained in () time as well as in the case where the multigrid method is used as a solver. Overview of Multigrid. Backslash \ is the build-in direct solver of MATLAB. In this novel coding style of a multigrid numerical solver which converges in linear time. This eddy current solver is a unique capability of ML and utilizes the discrete nullspace of the operator in building the smoothers and grid hierarchy. Mar 5, 2013 · Implementation of a multigrid solver on a GPU for Stokes equations with strongly variable viscosity based on Matlab and CUDA Liang Zheng , Huai Zhang [email protected] , […] , Taras Gerya , Matthew Knepley , David A Yuen , and Yaolin Shi [email protected] +3 -3 View all authors and affiliations AMG2023 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. Iterative methods generate a series of solutions from an initial guess, converging to a final result after several steps. We can apply coarse grid correction to the residual equation on \Omega^{2h} , which means that we need \Omega^{4h} for the correction step. AN ALGEBRAIC EQUATION OF TWO POINT BOUNDARY VALUE PROBLEMS We consider the discretization of Poisson equation with homogenous Dirichlet Here is a small comparison of Geometric Multigrid (GMG) and Algebraic Multigrid (AMG): Here is another test case: solving the linear system generated by FEM using uniform triangular element (using MATLAB code in this repo to generate data): Reference: Feb 19, 2019 · fd1d_bvp, a MATLAB code which applies the finite difference method to a two point boundary value problem in one spatial dimension. 引入一个例子考虑到用数学符号来描述过于抽象，因… ML contains a variety of parallel multigrid schemes: smoothed aggregation; FAS nonlinear multigrid; a special algebraic multigrid for the eddy current approximations to Maxwell’s equations. Contribute to alecjacobson/multigrid development by creating an account on GitHub. Download and share free MATLAB code, including functions, models, apps, support packages and toolboxes. Published with MATLAB® 7. m (CSE) Sets up a 1d Poisson test problem and solves it by multigrid. x = amg(A,b) implements algebraic multigrid solver. Combinatorial Multigrid is a linear system solver for symmetric diagonally dominant systems. implicit euler for time and central differencing for space. The idea is that we consider a problem on different refinement levels and use solutions on coarser levels to improve upon solutions on finer levels. The key new ingredients are the (rectangular !) matrices R and I that change grids: 1. On multigrid-CG for efficient topology optimization. Using the variable solver, choose the solver between MATLAB default solver, conjugate gradient method, gauss seidel and multi-grid method (with Gauss seidel iterations). This method is called Newton-multigrid and can be very effective. Multigrid methods are tremendously successful solvers for matrices arising from non-oscillatory PDE problems. Modern GPUs enjoy a laudable perfor- ----- >> Elastohydrodynamic Lubrication Point Contact Solver for MATLAB ----- Based on the theory presented in Venner and Lubrecht's book "Multilevel Methods in Lubrication", with the following modifications: - Fourier Transforms are used to calculate deformation instead of the Multi-Level Multi-Integration mentioned in the book - Parallel line solvers are provided. The method is however purely algebraic and may be tested on any problem. Skip to content. ipynb: Jupyter Notebook containing the Python version of the code which at this moment has bugs and it will be completed in future; report. Low-order-refined (LOR) is an alternative matrix-free methodology for solving these problems. 1. Examples of solvers that are not matrix-free. ods for nonlinear problems like the multigrid full approximation stor-age (FAS) scheme [4, 21]. Multigrid is especially successful for symmetric systems. While partial di er- Nov 13, 2020 · INTRODUCTION TO MULTIGRID METHODS LONG CHEN We give a short introduction to multigrid methods for solving the linear algebraic equa-tion arising from the discretization of Poisson equation in one dimension. Whereas ge ometric multigrid methods require a predetermined hierarchy of grids and discretizations, algebraic multigrid If you use PyAMG in your work, please consider using the following citation: @article{pyamg2023, author = {Nathan Bell and Luke N. Multigrid method is commonly used in reducing the iteration steps for solving the elliptic partial differential equation with the ill-conditioned matrix due to the saddle points in the matrix system coupling mass and momentum equations and strongly Feb 19, 2019 · FD1D_BVP, a MATLAB program which applies the finite difference method to a two point boundary value problem in one spatial dimension. Create a random symmetric sparse matrix A. The problem is when I increase the number of points i. As its name suggests, CMG borrows the structure and operations of multigrid algorithms. Algebraic Multigrid Methods Synonyms Algebraic Multigrid, AMG De nition Algebraic multigrid (AMG) methods are used to approximate solutions to (sparse) linear systems of equations using the multilevel strategy of relaxation and coarse-grid correction that are used in geometric multigrid (GMG) methods. I personally definitely want to meet the author and learn from him. Includes V, W, and F cycle See full list on mathworks. Poisson solvers can be used to solve a variety of physical problems either as a stand alone solver or as a part of another solver. 多重网格法的步骤. May 31, 2011 · I am trying to Solve Ax=B in MATLAB, where A is square matrix of size ~500,000 and B is the vector of same size. zip file here. UMFPACK is the solver behind the backslash command in MATLAB. pdf: Report submitted for the class for which this project was used iFEM is a MATLAB software package containing robust, efficient, and easy-following codes for the main building blocks of adaptive finite element methods on unstructured simplicial grids in both two and three dimensions. algebraic multigrid linear solver (https: physics matlab wave fem physics-simulation wave-equation 1d helmholtz-equation maxwell photonics optoelectronics cavity-simulators pwe dielectric maxwell-equations-solver photonic-mode-solver microcavity resonant-cavity 介绍代数多重网格的资料不少，但多数是泛（gen）泛（ben）而（bu）谈（dong），我根据自己的理解，对这一问题进行了推导及总结，如有不严谨的地方，欢迎指出。 1. The key new ingredients are the (rectangular!) matrices R and I that change grids: 1. zzvk wxnh pfjy ypg dhgqlxp zbmenltdf jdvg iziuo xjyup cqrey rpq bxacj aecbh fznumzb oygta