A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations (Lecture Notes in Statistics Vol.29) （2003. V, 194 p.）

A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations (Lecture Notes in Statistics Vol.29) （2003. V, 194 p.）

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  • 製本 Paperback:紙装版/ペーパーバック版／ページ数 194 p.
  • 商品コード 9783540003519

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the solution or its gradient. These new discretization techniques are promising approaches to overcome the severe problem of mesh-generation. Furthermore, the easy coupling of meshfree discretizations of continuous phenomena to dis­ crete particle models and the straightforward Lagrangian treatment of PDEs via these techniques make them very interesting from a practical as well as a theoretical point of view. Generally speaking, there are two different types of meshfree approaches; first, the classical particle methods [104, 105, 107, 108] and second, meshfree discretizations based on data fitting techniques [13, 39]. Traditional parti­ cle methods stem from physics applications like Boltzmann equations [3, 50] and are also of great interest in the mathematical modeling community since many applications nowadays require the use of molecular and atomistic mod­ els (for instance in semi-conductor design). Note however that these methods are Lagrangian methods; i. e. , they are based On a time-dependent formulation or conservation law and can be applied only within this context. In a particle method we use a discrete set of points to discretize the domain of interest and the solution at a certain time. The PDE is then transformed into equa­ tions of motion for the discrete particles such that the particles can be moved via these equations. After time discretization of the equations of motion we obtain a certain particle distribution for every time step.

Contents

1 Introduction.- 2 Partition of Unity Method.- 2.1 Construction of a Partition of Unity Space.- 2.2 Properties.- 2.3 Basic Convergence Theory.- 3 Treatment of Elliptic Equations.- 3.1 Galerkin Discretization.- 3.2 Boundary Conditions.- 3.3 Numerical Results.- 4 Multilevel Solution of the Resulting Linear System.- 4.1 Multilevel Iterative Solvers.- 4.2 Multilevel Partition of Unity Method.- 4.3 Numerical Results.- 5 Tree Partition of Unity Method.- 5.1 Single Level Cover Construction.- 5.2 Construction of a Sequence of PUM Spaces.- 5.3 Numerical Results.- 6 Parallelization and Implementational Details.- 6.1 Parallel Data Structures.- 6.2 Parallel Tree Partition of Unity Method.- 6.3 Numerical Results.- 7 Concluding Remarks.- Treatment of other Types of Equations.- A.1 Parabolic Equations.- A.2 Hyperbolic Equations.- Transformation of Keys.- Color Plates.- References.