A least‐squares smooth fitting for irregularly spaced data: Finite‐element approach using the cubic B-spline basis

Geophysics ◽  
1986 ◽  
Vol 51 (11) ◽  
pp. 2051-2066 ◽  
Author(s):  
Hiroshi Inoue
Keyword(s):  
Finite Element ◽  
Lateral Displacement ◽  
A Priori ◽  
Finite Element Approximation ◽  
Equation System ◽  
Smoothing Function ◽  
Element Approximation ◽  
Random Errors ◽  
B Spline ◽  
Second Derivatives

A new method of multivariate smooth fitting of scattered, noisy data using cubic B-splines was developed. An optimum smoothing function was defined to minimize the [Formula: see text] norm composed of the data residuals and the first and the second derivatives, which represent the total misfit, fluctuation, and roughness of the function, respectively. The function is approximated by a cubic B‐spline expansion with equispaced knots. The solution can be interpreted in three ways. From the stochastic viewpoint, it is the maximum‐likelihood estimate among the admissible functions under the a priori information that the first and second derivatives are zero everywhere due to random errors, i.e., white noise. From the physical viewpoint, it is the finite‐element approximation for a lateral displacement of a bar or a plate under tension which is pulled to the data points by springs. From a technical viewpoint, it is an improved spline‐fitting algorithm. The additional condition of minimizing the derivative norms stabilizes the linear equation system for the expansion coefficients.

A mixed finite-element method for elliptic operators with Wentzell boundary condition

2018 ◽  
Vol 40 (1) ◽  
pp. 87-108
Author(s):  
Eberhard Bänsch ◽  
Markus Gahn
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Weak Formulation ◽  
Wentzell Boundary Condition ◽  
Polyhedral Domain

Abstract In this paper we introduce and analyze a mixed finite-element approach for a coupled bulk-surface problem of second order with a Wentzell boundary condition. The problem is formulated on a domain with a curved smooth boundary. We introduce a mixed formulation that is equivalent to the usual weak formulation. Furthermore, optimal a priori error estimates between the exact solution and the finite-element approximation are derived. To this end, the curved domain is approximated by a polyhedral domain introducing an additional geometrical error that has to be bounded. A computational result confirms the theoretical findings.

scholarly journals AN IMPROVED FINITE ELEMENT APPROXIMATION AND SUPERCONVERGENCE FOR TEMPERATURE CONTROL PROBLEMS

2013 ◽  
Vol 18 (5) ◽  
pp. 631-640 ◽  
Author(s):  
Yuelong Tang
Keyword(s):  
Finite Element ◽  
Temperature Control ◽  
Piecewise Linear ◽  
A Priori ◽  
Finite Element Approximation ◽  
Linear Functions ◽  
Control Problems ◽  
Element Approximation ◽  
Adjoint State ◽  
Admissible Controls

In this paper, we consider an improved finite element approximation for temperature control problems, where the state and the adjoint state are discretized by piecewise linear functions while the control is not discretized directly. The numerical solution of the control is obtained by a projection of the adjoint state to the set of admissible controls. We derive a priori error estimates and superconvergence of second-order. Moreover, we present some numerical examples to illustrate our theoretical results.

scholarly journals AN H1 -GALERKIN MIXED FINITE ELEMENT APPROXIMATION OF A NONLOCAL HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (5) ◽  
pp. 643-653
Author(s):  
Fengxin Chen ◽  
Zhaojie Zhou
Keyword(s):  
Finite Element ◽  
Hyperbolic Equation ◽  
Mixed Finite Element ◽  
A Priori ◽  
Finite Element Approximation ◽  
Nonlinear Hyperbolic Equation ◽  
Element Approximation ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Priori Error Estimates

In this paper we investigate a semi-discrete H1 -Galerkin mixed finite element approximation of one kind of nolocal second order nonlinear hyperbolic equation, which is often used to describe vibration of an elastic string. A priori error estimates for the semi-discrete scheme are derived. A fully discrete scheme is constructed and one numerical example is given to verify the theoretical findings.

scholarly journals A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Chuanjun Chen ◽  
Wei Liu
Keyword(s):  
Finite Element ◽  
Piecewise Linear ◽  
A Priori ◽  
Grid Method ◽  
Finite Element Approximation ◽  
Nonlinear Parabolic Equations ◽  
Coarse Grid ◽  
Element Approximation ◽  
Fine Grid ◽  
Nonlinear Parabolic

A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid sizeH. The nonlinearities are expanded about the coarse grid solution on a fine gird of sizeh, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with theH1-normO(h+H2)which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfyh=O(H2). An example is also given to illustrate the theoretical results.

Mesh Quality and More Detailed Error Estimates of Finite Element Method

2017 ◽  
Vol 10 (2) ◽  
pp. 420-436
Author(s):  
Yunqing Huang ◽  
Liupeng Wang ◽  
Nianyu Yi
Keyword(s):  
Finite Element ◽  
Error Estimate ◽  
Error Estimates ◽  
A Priori ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Mesh Quality ◽  
Yield Information ◽  
Symmetry Structure

AbstractIn this paper, we study the role of mesh quality on the accuracy of linear finite element approximation. We derive a more detailed error estimate, which shows explicitly how the shape and size of elements, and symmetry structure of mesh effect on the error of numerical approximation. Two computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the mesh. In compare with the standard a priori error estimates, which only yield information on the asymptotic error behaviour in a global sense, our proposed error estimate considers the effect of local element geometry properties, and is thus more accurate. Under certain conditions, the traditional error estimates and supercovergence results can be derived from the proposed error estimate. Moreover, the estimators Ge and Gv are computable and thus can be used for predicting the variation of errors. Numerical tests are presented to illustrate the performance of the proposed parameters Ge and Gv.

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