Higher-Order Finite Element Methods

Šolín, Pavel; Segeth, Karel; Doležel, Ivo

doi:10.1201/9780203488041

Cited by 317 publications

(270 citation statements)

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“…This procedure has been used in previous work (e.g. [5,11]). For the onedimensional case, the use of Lobatto polynomials as shape functions can simplify significantly the solution of the discrete minimization problem.…”

Section: Recovery Of the Projection-based Interpolant In The One-dimementioning

confidence: 98%

“…Recently, Solin et al [11] proposed an extra refinement criteria to account for the actual magnitude of the projection-based error appearing in the coarse solution to identify elements of high error magnitude, namely err iel > 10err ave (12) where err ave is the average element projection-based error magnitude. This criteria will only be activated when a mesh contains a few elements of very high error decrease, which will distort the criteria given by (11), and may result in only a few elements being refined. Last line of (10) shows that the error decrease err iel coincides with the estimated error magnitude err iel .…”

Section: A Projection-based Interpolation Technique and The Alternatimentioning

confidence: 99%

“…Interested readers can refer to Deeks and Wolf [17] for a complete development of the scaled boundary finite element method and to Vu and Deeks [4] for a complete development of higher order scaled boundary finite element methods. The hierarchical Lobatto shape functions [11,18] are used due to their ability to produce accurate solutions for high polynomial order [4]. The scaled boundary method introduces a normalised radial coordinate system termed the scaled boundary coordinate system (ξ, s), which is illustrated for a typical bounded domain enclosed between two similar boundaries in Fig.…”

Section: Hierarchical Higher Order Elements In the Scaled Boundary Fimentioning

confidence: 99%

“…The reference solution and the projection-based interpolation technique have been adopted widely in the finite element method by various researchers to formulate refinement criteria and perform error estimation in the adaptivity process for various h-, p-, and hp-versions [5,[9][10][11][12]. Attempts have been made to quantify the error in various quantities of interest such as the average value of the solution over a specified subdomain and point-wise quantities of interest to form goal-oriented approaches for each of these adaptivity types [13][14][15].…”

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confidence: 99%

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A p-adaptive scaled boundary finite element method based on maximization of the error decrease rate

Deeks

2007

Comput Mech

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This study enhances the classical energy norm based adaptive procedure by introducing new refinement criteria, based on the projection-based interpolation technique and the steepest descent method, to drive mesh refinement for the scaled boundary finite element method. The technique is applied to p-adaptivity in this paper, but extension to h-and hp-adaptivity is straightforward. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh, is used to represent the unknown exact solution. In the new adaptive approach, a projection-based interpolation technique is developed for the 2D scaled boundary finite element method. New refinement criteria are proposed. The optimum mesh is assumed to be obtained by maximizing the decrease rate of the projectionbased interpolation error appearing in the current solution. This refinement strategy can be interpreted as applying the minimisation steepest descent method. Numerical studies show the new approach out-performs the conventional approach.

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Section: Recovery Of the Projection-based Interpolant In The One-dimementioning

confidence: 98%

Section: A Projection-based Interpolation Technique and The Alternatimentioning

confidence: 99%

Section: Hierarchical Higher Order Elements In the Scaled Boundary Fimentioning

confidence: 99%

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confidence: 99%

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A p-adaptive scaled boundary finite element method based on maximization of the error decrease rate

Deeks

2007

Comput Mech

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“…In the case of smooth solutions u in parts of the domain , spectral methods, [23], and finite elements of high order ( p-version), see e.g. [30], [31], [32], [9], and the references therein, have become more popular for twenty years. For the h-version of the FEM, the polynomial degree p of the shape functions on the elements is kept constant and the mesh-size h is decreased.…”

Section: Introductionmentioning

confidence: 99%

New shape functions for triangular p-FEM using integrated Jacobi polynomials

Beuchler

Schöberl

2006

Numer. Math.

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In this paper, the second order boundary value problem −∇ · (A(x, y) ∇u) = f is discretized by the Finite Element Method using piecewise polynomial functions of degree p on a triangular mesh. On the reference element, we define integrated Jacobi polynomials as interior ansatz functions. If A is a constant function on each triangle and each triangle has straight edges, we prove that the element stiffness matrix has not more than 25 2 p 2 nonzero matrix entries. An application for preconditioning is given. Numerical examples show the advantages of the proposed basis.

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