Some mixed finite element methods for biharmonic equation

A general spatial interpolation method for tidal properties has been developed by solving a partial differential equation with a combination of different orders of harmonic operators using a mixed finite element method. Numerically, the equation is solved implicitly without iteration on an unstructured triangular mesh grid. The paper demonstrates the performance of the method for tidal property fields with different characteristics, boundary complexity, number of input data points, and data point distribution. The method has been successfully applied under several different tidal environments, including an idealized distribution in a square basin, coamplitude and cophase lines in the Taylor semi-infiite rotating channel, and tide coamplitude and cophase lines in the Bohai Sea and Chesapeake Bay. Compared to Laplace's equation that NOAA/NOS currently uses for interpolation in hydrographic and oceanographic applications, the multiple-order harmonic equation method eliminates the problem of singularities at data points, and produces interpolation results with better accuracy and precision.

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“…To directly solve a higher order harmonic equation, we employ a mixed finite element method [15][16][17] to transform Equation (4) …”

Section: Mixed Finite Element Methodsmentioning

confidence: 99%

Spatial Interpolation of Tidal Data Using a Multiple-Order Harmonic Equation for Unstructured Grids

Shi

Heß²,

Myers³

2013

IJG

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“…There have been many works on finite element methods for the fourth order partial differential equations (PDEs, for short), of course, containing the bi-harmonic equation as well, such as in [2,9,17,19] and so on. The problems described by bi-harmonic equations arise from fluid mechanics and solid mechanics, such as bending of elastic plates.…”

Section: Introductionmentioning

confidence: 99%

“…There has been much research about mixed finite element methods for the 4th order PDEs, for example, Ciarlet-Raviart elements, Herrmann-Miyoshi elements, HellanHerrmann-Johnson elements. More details can be found in [2,9,13,17,19] and the references cited therein. Optimal control problems governed by the fourth order PDEs also are encountered in many engineering applications.…”

Section: Introductionmentioning

confidence: 99%

Error Estimates of Mixed Finite Element Approximations for a Class of Fourth Order Elliptic Control Problems

Hou¹

2013

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we consider the error estimates of the numerical solutions of a class of fourth order linear-quadratic elliptic optimal control problems by using mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k ≥ 1). L 2 and L ∞ -error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for fourth order elliptic control problems.

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“…Therefore, we aim at replacing the second order derivative in the thin-plate spline formulation with a first order derivative by using a H 1 -conforming finite element method. This idea has been exploited to solve the biharmonic equation and to discretise the thin-plate spline [3,4,5,7,9,11,12,13,15].…”

Section: Introductionmentioning

confidence: 99%

“…Our new formulation is obtained by introducing an auxiliary variable σ = ∇u such that the minimisation problem (1) is rewritten as [3,9] min…”

Section: Introductionmentioning

confidence: 99%

A mixed finite element discretisation of thin-plate splines

Lamichhane¹,

Roberts²,

Stals³

2011

ANZIAMJ

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Thin-plate splines are a well established technique for the interpolation and smoothing of scattered data. However, the traditional formulation of the method leads to large, dense and often ill-conditioned matrices, which reduces its applicability in practice. We present a new mixed finite element formulation based on the ideas behind the mortar finite element methods. The resulting system of equations is sparse and positive definite, and its size depends only on the number of finite elements not the number of data points.

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Some mixed finite element methods for biharmonic equation

Cited by 42 publications

References 13 publications

Spatial Interpolation of Tidal Data Using a Multiple-Order Harmonic Equation for Unstructured Grids

Spatial Interpolation of Tidal Data Using a Multiple-Order Harmonic Equation for Unstructured Grids

Error Estimates of Mixed Finite Element Approximations for a Class of Fourth Order Elliptic Control Problems

A mixed finite element discretisation of thin-plate splines

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