@phdthesis{oai:niigata-u.repo.nii.ac.jp:02000948,
 author = {Galindo, Shirley Mae Alfarero},
 month = {2023-04-24, 2023-04-24},
 note = {The Lagrange interpolation is one of the most used interpolation types to approximate functions. Its interpolation error has been estimated under various norms and is a widely explored topic in numerical analysis. Error estimation for the approximation of functions greatly affects the development of numerical solutions to partial differential equations. For example, the maximum norm of the interpolation error influences the discretization error of the finite element method (FEM) solution. In this research, we consider the L ∞-norm estimation for the linear Lagrange interpolation over a triangle element K by using the H2-seminorm of the objective function, that is, to obtain the explicit estimate of the constant satisfying ∥u−ΠLu∥∞,K ≤ CL(K)|u|2,K, ∀u ∈ H2(K). Here, CL(K) is the interpolation error constant to be evaluated explicitly. Waldron (1998) estimated the maximum norm of the Lagrange interpolation error in terms of the maximum of the second derivative of the objective function, i.e., ∥u(2)∥∞,K. Since the H2-seminorm of the function requires less function regularity than ∥u(2)∥∞,K norm, the result of this research has more application. For triangle element K of general shape, a formula to give an estimate of the interpolation error constant CL(K) is obtained through theoretical analysis. The theoretical estimation leads to a raw bound that works well for triangle of arbitrary shapes. Particularly, our analysis tells that the value of CL(K) can be very large and tend to ∞ if the triangle element tends to degenerate to a 1D segment. An algorithm for the optimal estimation for CL(K) is also proposed, which extends the technique of eigenvalue estimation by Liu [17]. The objective problem is converted to a quadratic minimization problem with maximum norm constraint. To overcome the difficulty caused by the maximum norm in the constraint condition, a novel method is proposed by utilizing the orthogonality property of the interpolation associated to the Fujino-Morley FEM space and the convex-hull property of the Bernstein representation of functions in the FEM space. Specifically, for a unit right isosceles triangle K, it has been shown by rigorous computation that 0.40432 ≤ CL(K) ≤ 0.41596. Also, the maximum norm error estimation of the Lagrange interpolation has been used to estimate the local maximum error of the FEM solution to the Poisson boundary value problem. That is, given the subdomain Ω' ⊆ Ω, an estimator ξ is desired which satisfies ∥u−uh∥∞,Ω' ≤ ξ (u : exact solution, uh : FEM solution). To compute the above estimator, Fujita's method of pointwise estimation of the boundary value solution is employed, as well as the interpolation error estimation for the Lagrange interpolation. The code is shared at https://ganjin.online/shirley/InterpolationErrorEstimate. Main results of this dissertation can be found in [13]., 新大院博(理)第479号},
 school = {新潟大学, Niigata University},
 title = {Maximum norm error estimation for the finite element solution to partial differential equations},
 year = {}
}