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A Moment Method on Inverse Problems for the Heat Equation
Published in G Nakamura, S Saitoh, J K Seo, M Yamamoto, Inverse problems and related topics, 2019
Mishio Kawashita, Yaroslav Kurylev, Hideo Soga
and χ(t) is a (arbitrary) fixed C∞ function satisfying 0≤χ(t)≤1 in R, χ(t) = 1 for t ≥ 1 and χ(t) = 0 for t ≤ 1/2. The function p(x′) in (1.1) is the boundary value of a harmonic polynomial p(x) (i.e. Δp = 0). By H Pm we denote the set of all harmonic polynomials of degree ≤ m (m = 0,1,2,⋯). For any p,q∈HP∞(=∪m=0∞HPm) we put () Φρ(t;p,q)=∫Ωρ(x)up(t,x)q(x)¯dx,Mρ(p,q)=∫Ωρ(x)p(x)q(x)¯dx,
Topology optimization based on the high-order numerical manifold method by implementing a four-node quadrilateral element
Published in Engineering Optimization, 2023
Mohammad Kamalodini, Saleh Hamzehei-Javaran, Saeed Shojaee
LD is one of the most challenging issues in the NMM. This problem was first observed by Babuška and Melenk (1997). They found that in the global approximation, the number of unknowns is greater than the number of shape functions after the Dirichlet boundary conditions are precisely satisfied. Therefore, the unknowns are linearly dependent and the global matrix is rank deficient. Since the global approximation in NMM is obtained by multiplying the PU and the local approximation function, when the PU and the local approximation function are selected as a polynomial, the LD problem arises and the global matrix will be singular. The two functions that are most considered for the local approximation function are complete and harmonic polynomial functions. The complete polynomial function is as follows: in which q is the order of the local approximation function. The harmonic polynomial function for q = 0–3 is expressed explicitly as follows:
Conforming and nonconforming harmonic finite elements
Published in Applicable Analysis, 2020
Tatyana Sorokina, Shangyou Zhang
For any harmonic function on a triangle K, there is a harmonic polynomial such that
Harmonic interpolation of Hermite type based on Radon projections with constant distances
Published in Applicable Analysis, 2019
By our settings, in polar coordinates, we can write The series at the right-hand side converges uniformly on . Hence, from Equation (11), we have where . Since and for all , we get It follows from the Weierstrass test that the series of derivatives converges uniformly on for every . Hence, from Equation (18) and [17, Corollary 3.7.3], we obtain the formula for the derivative of with respect to θ, Next we rewrite , where the are pairwise distinct and the 's are positive integers. Since is a harmonic polynomial of total degree at most n, it is of the form It follows from Equation (11) that We recall interpolation conditions for , that is, Using Equations (19) and (21), we conclude from Equation (22) that We can regard Equation (23) as a system of 2n+1 linear equations where are unknowns. We will find the solution by using the Cramer rule. The determinant of the coefficient matrix equals On the other hand, the determinant of the matrix formed by replacing the column vector of the coefficient matrix corresponding to by the column vector formed by the right-hand side of Equation (23) is equal to It follows that Combining Equations (20) and (24) we obtain the desired formula. The proof is complete.