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By Yarmukhamedov Sh.

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We will now analyze more in detail how to build C 0 -conforming ﬁnite element spaces using Lagrangian ﬁnite elements. First of all, we note that to have non trivial C 0 -conforming spaces we need that r ≥ 1. In a Lagrangian ﬁnite element of degree r ≥ 1 we have that P (K) ⊂ Pr (K) and we can identify the local degrees of freedom through a set of points Ni ∈ K, i = 1, . . , nl , called nodes, by setting σi (p) = p(Ni ). The corresponding shape functions −1 and degrees of freedom on the current element K are given by φK,i = φi ◦TK and σK,i (p) = p(NK,i ), where NK,i = TK (Ni ) is the ith node of element K.

Let di indicate the approximation of the ﬁrst derivative of u at xi = ih with i ∈ Z and h > 0. 46) where the coeﬃcients a0 , a1 , b0 , b1 are chosen so that we obtain the maximal order of accuracy. Check the result by approximating the derivative of f (x) = sin(2πx) for x ∈ (0, 1). 4. Numerical approximation To ﬁnd the coeﬃcients that maximize the accuracy we compute the truncation error τi (u). e. if we set di = Du(xi ) = Dui . In our case τi (u) = a1 (Dui−1 + Dui+1 ) + a0 Dui − b1 (ui+1 − ui−1 ) − b0 ui .

31). 31) on the whole interval [0, 1], exploiting the periodicity. Program 4 returns in the vector dfdxp the approximate values of the pth derivative of a periodic function, given in a string or the inline function fun, at nh+1 equidistant nodes on the interval (xspan(1),xspan(2)). The discretization nodes are given in the output vector x, assuming x(1) = xspan(1) and x(nh+1)=xspan(2). The approximation method of Program 4 is of the general form Dui 1 hp i+N c k uk , k=i−N where the values of the coeﬃcients {ck } must be given in input in the vector coeff.