| The finite element method (FEM) has been widely used in computing partial difference functions. But with the net denser, the condition number of the linear system is increasing rapidly, which makes trouble in application. Therefore, it is necessary to find an effective precondition method to reduce the condition number.About Nonconforming FEM, as the natural finite element space is not nested, which bring new difficulty for its precondition. In this paper, we take the Poisson function as the model and change it to a saddle-point problem by introduce Lagrange multiplier method. In that way we change the unnested space to a nested one, which give us a way to construct wavelets basis.Consider the following Poisson function:where Ω is a polygonal region in R~2. f ∈ L~2(Ω) is a given function.Make rectangle subdivision on Ω, marks where K_i is rectanglecell, N is the number of the cells.The Lagrange multiplier method for (1)(2) is: find u ∈ H~1(Ω), λ ∈ M such thatwhere M is Lagrange multiplier space, that belong to L~2 on each edge of T_h,On each edge e of T/,, b(-, ?) is definite as:b(v,\) = ±r f[v}eXds (6)lel Jewhere [v]e = Vin — Vout, is the jump of v on e. The trial space of (3) (4) is defined as:Hh = {vh;vh\KeQl(K), VKeTh)where Q1(K) is rotated bilinear function space, that means on each K have the form a + bx\ + CX2 + d{x\ — rc|).Mh = {A/,;A/, is constant on each edge of Th}(3)(4) change to the problem: find ?/, € Hh, A/, G M/i such thata(uh, vh) + 6(?fc, Ah) =< /, vh >, Vvh G Hh (7)0, V?heMh (8)Lemma 1. For all the i;/, C ^, there exist a positive constant a > 0 independent of Vh, such thato(va,wa) > oc\vh\\. (9)Lemma 2.(LBB Condition) For all the (%,///,) G Hh x M/,, there exist a positive constant /3 > 0 , such thatinf sup ? ,^'f^,, >/?;(10)From Lemma 1 and Lemma 2 we can get:Theorem 1. Problem (7)(8)has the only solution (w/i,A/,) G Hh x M^, and has the error estimates:||u-ufc||o + ||A- Aa||m < C{ inf ||?-vA||o+ inf ||A-^||m}-Let fi = [0,1] x [0,1], we make square or 2 x 1 rectangle subdivision on it. Each subdivision is made on the former subdivision.$° is the vector composed by the basis of the 1st subdivision . $n is the vector composed by the basis of the (n+1) times subdivision. It's easy to know $° has 4 elements and $" has 2n+1 elements.Let V° is the space spread by $°, Vn is the space spread by $n we will proof:v° cv1 cv2c---cvn---.Prom (11) we can getvn = v° ? w1 ? w2 ? ? ? ? e wn. (12)Where Wk is the supplementary space of V*-1 in Vk, that is Vk — Vk~l ? Wk.In the unit square, we can use the mean values over the four edges as degrees of freedom and get four bases in the Ql form:OO0 O , 9 9.1 O / 9 O\satisfy/1 z= j 0 i # j,? ?? _ 1 0 Q 4According to Figure 1:Figure 1Divide the unit square to two 2x1 rectangles, by the same way to choosedegrees of freedom we can get 8 bases on the two rectangles: .11 9 17 6 12.n 3 12 16 12+xx1 7 6 122 12 4 123 17 18 122 14 12 28 12' = +XX13 7 18 1219r =8 i.2iAccording to Figure 2:1,2Figure 2)lfcLet ^'* corresponds to the edge e)lfc, k = 1,2;j = 1,2,3,4. such thatAccording to the former: $° = ($>, 0°, ^, 0°<^l (A1'1 A1'1 A1'1 A1'1 ^.i.2 A1'2 A1'2 Al<2\T $ = (4V ,??>2 ,{,ip2,4>l,ijl)T, then we have the relation1 \ (15)WhereG =1 0 0 0-1000 0 0 1 0 0 0-101 -1 -1 -1 -1110 0 0 10 10 0Let(16)the nesting matrix of $fc~1 and ^fe on $fe, we haveGkBy the above relation we can get the transition Cn between the wavelets basis T and the natural basis $n.(17)By the preconditioner Cn we can change the linear system under $" to the new system under wavelets basis.'CnAnCnT CnBn\ (Cn (A / An Bn\ fcjo ) v o i) \BnL o j v o /;(18)By computation we have proved that the condition number after precondition is obviously improved. (You can see the tabulations in §4 for more details)In conclusion, the multiscale basis method provided in this paper is an effective precondition method for nonconforming Lagrange multiplier method with basis in Ql form. Its validity is proved by computation. The theoretic proof of this method still need us for further research.
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