Geometric multigrid preconditioners for dpg systems in. The hhg framework is a carefully designed and implemented high performance finite element geometric multigrid software package. High level implementation of geometric multigrid solvers. Instead, it exploits a sequence of linear systems created by a sequence of refined meshes. Based on a finite element discretization with continuous finite elements, the taylorgalerkin approach is globally mass conserving. High level implementation of geometric multigrid solvers for finite element problems. List of finite element software packages wikipedia. Fast, robust and e cient multigrid solvers are a key numer. This is a list of software packages that implement the finite element method for solving partial. Ieee transactions on pattern analysis and machine intelligence. The purpose of finite element analysis fea software is to reduce the number of prototypes and experiments that have to be run when designing, optimizing, or controlling a device or process. A parallel geometric multigrid method for finite elements on. Scalable multigrid methods for immersed finite element methods. The advantage of geometric multigrid over algebraic multigrid is that the former should perform better for nonlinear problems since nonlinearities in the system are carried down to the coarse levels.
Easy geometric multigrid is a library of easy to use and understand geometric multigrid components for education and evaluation. What is the best iteration method used in finite element. Introduction outline 1 introduction 2 relation between finite element and mimetic finite di erence 3 geometric multigrid methods 4 conclusions and future work x. The resulting discrete problem does not fall into the standard variational framework for analyzing multigrid methods since the bilinear forms on different grid levels are not suitably related to each other. Highorder finiteelement package using hexahedral elements. Abstractparallel finite element algorithms based on object. Highorder finite element package using hexahedral elements. E cient finite element geometric multigrid solvers for.
Efficient finite element geometric multigrid solvers for unstructured. Since camellias adaptive mesh hierarchy provides us with rich geometric information, we focus on hp geometric multigrid preconditioners with additive schwarz smoothers of minimal or small overlap. We develop a gpu parallelization of a matrixfree geometric multigrid iterative. We illustrate its use, and investigate the performance of the resulting method, in the development of a geometric multigrid solver for a mimetic finite element discretisation of the. We usemaximal independentsets miss as a mechanism to automatically coarsen unstructured grids.
On the software and, more importantly in the scope of this paper, the algorithmic. There are many methods like multigrid and conjugate gradient methods. Multigrid is a very special solver which differs slightly in setting up a corresponding solver tree. At the theoretical level, shaidurov justifies the rate of convergence of various multigrid algorithms for selfadjoint and nonselfadjoint problems, positive definite and indefinite problems, and singular and spectral problems. A flexible, parallel, adaptive geometric multigrid method for fem. Parallel multigrid solver for 3d unstructured finite element. The implementation of efficient multigrid preconditioners for elliptic partial differential equations pdes is a challenge due to the complexity of the resulting algorithms and corresponding computer code.
Pdf multigrid solvers for immersed finite element methods. These discretizations are described in detail in the lecture notes of numerical mathematics iii. Hhg combines the flexibility of unstructured finite element meshes. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides meshindependent and cut element independent convergence rates. Which is the best method to iterate the required v. Nov 26, 2019 this contribution develops a geometric multigrid preconditioner that enables iterative solutions for higherorder immersed finite element methods at a computational cost that is linear with the number of degrees of freedom. Manually setup the geometric multigrid solver knowledge base. Clemson university tigerprints all dissertations dissertations december 2019 a parallel geometric multigrid method for adaptive finite elements thomas conrad. We consider the flux limited taylorgalerkin approach as a suitable compromise that is easily realized in finite element software. It supports trilinear finite element discretizations constructed using octees. When working with unstructured grids it is really difficult to define a series of meshes for geometric multigrid so algebraic multigrid is usually used instead.
The hyteg finite element software framework for scalable multigrid solvers. Geometric multigrid fas cfdwiki, the free cfd reference. The discretized equations are evaluated on every level. General fea software what does finite element analysis software bring. The idea is to adapt the general convergence theory for geometric multigrid to our speci. Applications in atmospheric modelling mitchell, lawrence muller, eike hermann. Algebraic multigrid methods for higherorder finite element. This article presents matrixfree finite element techniques for efficiently solving partial differential equations on modern manycore processors, such as graphics cards. The hierarchical hybrid grids hhg software framework 3, 17 is designed to close this gap between finite element flexibility and geometric multigrid performance by using a compromise between. High level implementation of geometric multigrid solvers for. The weak formulation of the pde is expressed in unified form language ufl and the lower pyop2 abstraction layer allows the manual design of computational kernels for a bespoke geometric multigrid preconditioner.
I didnt realize they had intrinsic geometric multigrid support in deal. To clarify, geometric multigrid actually discretizes the problem on a series of grids whereas algebraic multigrid simply works with the linear system. This is a list of software packages that implement the finite element method for solving partial differential equations. Finite element analysis software uses iteration to find displacement vector. Multigrid solvers for immersed finite element methods and. An essential aspect of finite element methods and isogeometric analysis is the computation of the solution to a system of equations. A massively parallel multigrid method for finite elements. We compare the performance of this preconditioner to a singlelevel method and hypres boomeramg algorithm. The code is a testbed for geometric multigrid approaches for high order discretizations. The stiffness, geometric stiffness, and mass matrices for an element are normally derived in the finite element analysis by substituting the assumed displacement field into the principle of virtual work. This sequence of meshes generates a sequence of finite element spaces x1. Multigrid methods can be applied in combination with any of the common discretization techniques. E cient finite element geometric multigrid solvers for unstructured grids on gpus markus geveler, dirk ribbrock, dominik g oddeke, peter zajac, and stefan turek applied mathematics ls3, tu dortmund, germany, markus. For example, the finite element method may be recast as a multigrid method.
Such methods were implemented for instance in the software libraries. The multigrid algorithm starts with the initial physics controlled mesh or user defined mesh and automatically builds a series of coarser meshes. Multigrid solvers for immersed finite element methods and immersed. Multigrid methods for finite elements combines two rapidly developing fields. Efficient multigrid solvers for mixed finite element discretisations in nwp models author colin cotter, david ham, lawrence mitchell, eike hermann muller, robert scheichl.
In this article, we present a parallel geometric multigrid algorithm for solving variablecoefficient elliptic partial differential equations on the unit box with dirichlet or neumann boundary conditions using highly nonuniform, octreebased, conforming finite element discretizations. Numerical integration is technically convenient and used routinely as a device in the finite element method. A finite element multigridframework to solve the sea ice. The docs appear to still say scary things like the interface of this class is still very clumsy. Finite element multigrid framework for mimetic finite.
For sophisticated mixed finite element discretisations on unstructured grids an efficient implementation can be very time consuming and requires the programmer to have indepth knowledge. International journal of parallel, emergent and distributed systems. For applying the p1 finite element method, the grid has to be decomposed into triangles. In this article, we present a parallel geometric multigrid algorithm for solving ellip. In this paper, we discuss the extension of the firedrake framework to support the development of geometric multigrid solvers for finite element problems. High level implementation of geometric multigrid solvers for finite. In these cases, multigrid methods are among the fastest solution techniques known today. Finite element analysis fea software comsol multiphysics. Multigrid methods are solvers for linear system of equations that arise, e. Yes, yes, yes up to 147k processes, test for 4k processes and geometric multigrid for 147k, strong and weak scaling, no, yes, demonstrated. Both memory and time costs therefore motivate the present work, an exploration of iterative solvers in the context of poisson and stokes problems. An object oriented parallel finite element scheme for computations of. Their big limitation last time i checked was geometric element type.
Efficient multigrid solvers for mixed finite element. Multigrid methods for a mixed finite element method of the darcyforchheimer model jian huang long chen hongxing rui received. Multigrid methods for a mixed finite element method of the. For sophisticated finite element discretisations on unstructured grids an efficient implementation can be very time consuming and requires the programmer to have indepth knowledge of the. The algebraic multigrid method presented is designed for. Geometric multigrid, finite element method, linear octrees, adaptive meshes. Fem software are being explored stronger smoothers are still an issue spai, ilu complete geometric. Geometric multigrid or fas in geometric multigrid a hierarchy of meshes is generated. Multigrid methods for finite elements mathematics and its. List of finite element software packages explained this is a list of software packages that implement the finite element method for solving partial differential equations. A multigrid solver does not rely on just one linear system on one mesh.
It builds on the abstractions introduced in the fenics project 16, 22 to present a highlevel, automated, problem solving environment. Illconditioning of the system matrix is a wellknown complication in immersed finite element methods and trimmed isogeometric analysis. Have you read some literature about the geometric multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides meshindependent and cutelementindependent convergence rates. The hyteg finiteelement software framework for scalable. Firedrake 10 is a python system for the solution of partial differential equations by the finite element method. Actually, this vcycle solver is a purely algebraic solver amg. The geometric multigrid solver accelerates the convergence of the iterative solver by solving the finite element problem on a series of meshes rather than a single one.
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