By P Wesseling

Multigrid equipment have constructed swiftly and are used as a robust instrument for the effective answer of elliptic and hyperbolic equations. this article presents an creation to multigrid equipment for partial differential equations, with purposes to sensible circulate difficulties.

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**Example text**

1. Let (X, +) be a topological group, and A, B ⊂ X two sets of the second category with the Baire property. Then int (A + B) = ∅ . 3 Added in the 2nd edition by K. Baron. 9. Theorem of S. Piccard 43 Proof. X itself is of the second category, since it contains sets of the second category. Then every non-empty open subset of X is of the second category. In fact, suppose that G ⊂ X is non-empty, open and of the ﬁrst category. Fix an a ∈ G. 5) are homeomorphisms and x ∈ G − a + x. Thus X is of the ﬁrst category at every point x ∈ X.

2) In fact, (H ∪ C) \ D = [H ∪ R ∪ (G \ H)] ∩ (P ∩ R ) = [H ∪ R ∪ (G ∩ H )] ∩ (P ∪ R) = (H ∪ R ∪ G ) ∩ (H ∪ R ∪ H ) ∩ (P ∪ R) . Now, H ∪ R ∪ H = X, and H ∪ R ∪ G = R ∪ G , since H = int G ⊂ G . Hence (H ∪C)\D = (R∪G )∩(P ∪R) = (R∩P )∪R∪(G ∩P )∪(G ∩R) = (G ∩P )∪R = A since R ∩ P ⊂ R and G ∩ R ⊂ R. 2). Now, the set H is open, D ⊂ P is of the ﬁrst category, and G \ H is a closed set without inner points, and so it is nowhere dense, and hence of the ﬁrst category. 2) that A has the Baire property. 2.

We do not presuppose any of those approaches. At any case the reader should be familiar with the following theorem, which we quote here without proof. 1. For any set A ⊂ RN the following conditions are equivalent: (i) A ∈ L. (ii) For every ε > 0 there exist open sets G and U such that A ⊂ G , G \ A ⊂ U , m(U ) < ε . (iii) There exist sets H ∈ Gδ and B ∈ L such that A ⊂ H , A = H \ B , m(B) = 0 . (iv) For every ε > 0 there exist a closed set F and an open set V such that F ⊂ A , A \ F ⊂ V , m(V ) < ε .