An Aditive Schwarz Method for the P Version Finite Element Method
An Aditive Schwarz Method for the P Version Finite Element Method
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We show that the condition number of the ASM iteration operator is bounded by a constant independent of p. The proof is similar to the one in Dryja and Widlund [4] and is based on Lions' partitioning lemma.
This paper is organized as follows. In Section 2, we define a model problem and introduce its discretization with the p-version finite element method. In Section 3, we review the basic framework of the ASM and apply it to our model problem with square elements in two dimensions. An ASM using... brick-shaped elements in three dimensions is considered in Section 4. For an example of a method similar to ours, but using the h-version finite element method, see Bramble et al. , [2].
2 The Model Problem.
We consider a model problem for linear, self adjoint, second order elliptic problems, on a bounded Lipschitz region Q. The discrete problem is given by the p-version finite element method. For simplicity, we first consider the following problem in R^: -Au = f in fi, u = on dCt .
The standard variational formulation of this problem is : Find u e V = H^iQ) such that a(u, v) = f{v), V uG F, where the bilinear form a(u, v) = / Vu • Vv dx Jn defines a semi-norm |w|//i(fi) = (a{u, u))^/'^ in H\Q), and a norm in V = H^{il).
What to read after An Aditive Schwarz Method for the P Version Finite Element Method?
You can find similar books in the "Read Also" column, or choose other free books by Luca F Pavarino to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
We show that the condition number of the ASM iteration operator is bounded by a constant independent of p. The proof is similar to the one in Dryja and Widlund [4] and is based on Lions' partitioning lemma.
This paper is organized as follows. In Section 2, we define a model problem and introduce its discretization with the p-version finite element method. In Section 3, we review the basic framework of the ASM and apply it to our model problem with square elements in two dimensions. An ASM using... brick-shaped elements in three dimensions is considered in Section 4. For an example of a method similar to ours, but using the h-version finite element method, see Bramble et al. , [2].
2 The Model Problem.
We consider a model problem for linear, self adjoint, second order elliptic problems, on a bounded Lipschitz region Q. The discrete problem is given by the p-version finite element method. For simplicity, we first consider the following problem in R^: -Au = f in fi, u = on dCt .
The standard variational formulation of this problem is : Find u e V = H^iQ) such that a(u, v) = f{v), V uG F, where the bilinear form a(u, v) = / Vu • Vv dx Jn defines a semi-norm |w|//i(fi) = (a{u, u))^/'^ in H\Q), and a norm in V = H^{il).
What to read after An Aditive Schwarz Method for the P Version Finite Element Method?
You can find similar books in the "Read Also" column, or choose other free books by Luca F Pavarino to read onlineMoreLess
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