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Well-posedness of the dual mixed formulation 83 We begin our course of mixed finite element methods by studying a vector-.
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Mixed Finite Element Methods

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Dauge, "Maxwell and Lame eigenvalues on polyhedra," Math. CO; However, Formula deserves another very important comment. Actually, we did not specify the norms adopted for x , y , f , and g. We had the right to do so, since in finite dimension, all norms are equivalent. Hence, the change of one norm with another would only result in a change of the numerical value of the constant c , but it would not change the basic fact that such a constant exists. However, in dealing with linear systems resulting from the discretization of a partial differential equation, we face a slightly different situation.

In fact, if we want to analyze the behavior of a given method when the mesh size becomes smaller and smaller, we must ideally consider a sequence of linear systems whose dimension increases and approaches infinity when the mesh size tends to zero. As it is well known and it can also be easily verified , the constants involved in the equivalence of different norms depend on the dimension of the space.

When considering a sequence of problems with increasing dimension, we have to take into account that n and m become unbounded. However, even if inequality holds with a constant c independent of h , it will not provide a good concept of stability unless the four norms are properly chosen Remark 1. This is going to be our next task. We start denoting by X , Y , F , and G , respectively, the spaces of vectors x , y , f , and g. Then, we assume what follows. Stability Definition.

In particular, to guarantee stability condition , we need to introduce two assumptions involving such matrices. Among the problems presented in Section 2 , this requirement is verified in practice only for the Stokes problem. Finally, we will briefly discuss more complicated problems, omitting the proofs for simplicity.

The basic assumption that we are going to use, throughout the whole section, deals with the matrix B. We assume the following: inf—sup Condition.

Remark 3. To better understand the meaning of , it might be useful to see when it fails. However, the injectivity of B T is not sufficient for the fulfillment of condition We will see in Proposition 1 that all these considerations on the particular matrix B in do extend to the general case. We now rewrite condition in different equivalent forms, which will also make clear the reason why it is called inf—sup condition. Let us see now the relationship of the inf—sup condition with a basic property of the matrix B. Proposition 4.

Assume that the inf—sup condition holds, and let y be any vector in Y. Assume conversely that holds. Remark 6.

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We point out once again cf. Remark 1 that the injectivity of B T is not sufficient for the fulfillment of the inf—sup condition. Additional relationships between the inf—sup and other properties of the matrix B will be presented later on in Section 3. As we will see in the sequel, the inf—sup condition is a necessary condition for having stability of problems of the general form In order to have sufficient conditions, we now introduce a further assumption on the matrix A.