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Adaptive Multigrid FE Methods -- An optimal way to solve PDEs

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##### Adaptive Multigrid FE Methods -- An optimal way to solve PDEs

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**Adaptive Multigrid FE Methods -- An optimal way to solve**PDEs Zhiming Chen Institute of Computational Mathematics Chinese Academy of Sciences Beijing 100080**Adaptive Concept**Refine: more nodes around singularities Coarsen: less nodes in smooth region**The adaptive method finds the solution of given tolerance**on a self-generated mesh according to the properties of the solution (singularities,oscillations). • An “optimal mesh” is the mesh on which the error is approximately the same on each element. This motivates the error equi-distribution strategy. • The adaptive FEM based on a posteriori error estimates provides a systematic way to refine or coarsen the mesh according to the local a posteriori error estimators on each element.**A Linear Elliptic Problem**Elliptic problem with piecewise constant coefficients: Variational problem:**Discrete Problem: find such that**A priori error estimate:**A posteriori error estimate (Babuska & Miller, 1987)**The error indicator where**Adaptive Algorithm**Solve → Estimate → Refine/Coarsen Error equi-distribution strategy where tolerance, constant , number of elements in**Numerical Experiments**where and Exact solution (Kellogg)**FEM with uniform mesh**128x128 mesh: 512x512 mesh: 1024x1024 mesh: Convergence rate: A priori error analysis implies that one must introduce nodes in each space direction to bring the energy error under 0.1.**The surface plot of the relative error**The maximum of the relative error is 0.2368.**The adaptive mesh of 2673 nodes. The energy error is**0.07451.**The surface plot of the adaptive solution and the**relative error . The maximum of the relative error is0.0188.**Definition**Let be the sequence of FE solutions generated by the adaptive algorithm. The meshes and the associated numerical complexity are called quasi-optimal if are valid asymptotically. DOFs(k) is the number of degree of freedoms of the mesh .**Quasi-optimality of the estimators. The quasi-optimal decay**is indicated by the dotted line of slope –1/2.**Adaptive Multigrid Method**◆ Local relaxation: Gauss-Seidel relaxation performed only on new nodes and their immediate neighboring nodes ◆ Each multigrid iteration requires only O(N) number of operations ◆ Theorem (Wu and Chen): We have ◆ Numerical Example**Applications**• Continuous casting problem Chen, Nochetto and Schmidt (2000) • Wave scattering by periodic structure Chen and Haijun Wu (2002) • Convection diffusion problem Chen and Guanghua Ji (2003)**例子：振荡铸钢速度系数**变化铸钢速度: 速度 v(t), 单元个数, 时间步长**A Linear Convection Diffusion Problem**Rotating Cylinder problem:**Convergence Rate**Epsilon=10e-3 Epsilon=10e-5