Corners Give Problems When Decoupling Fourth Order Equations Into Second Order Systems

Gerasimov, Tymofiy; Stylianou, Athanasios; Sweers, Guido

doi:10.1137/100806151

Cited by 19 publications

(14 citation statements)

References 11 publications

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“…The higher spatial gradient in the energy is treated using a Ciarlet–Raviart–Monk mixed formulation 13,14 with clamped boundary conditions u = − 1 on normal∂ normalΩ and ∂ u / ∂ n = 0 on normal∂ normalΩ and we note that on our convex domain this variational crime is only a misdemeanor. 15…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The higher spatial gradient in the energy is treated using a Ciarlet-Raviart-Monk mixed formulation 13,14 with clamped boundary conditions u = −1 on ∂Ω and ∂u/∂n = 0 on ∂Ω and we note that on our convex domain this variational crime is only a misdemeanor. 15 The initial condition is an approximation of the characteristic function of an ellipsoid with principle axes 0.7, 0.3, and 0.3. As seen in Figure 2, in the simulation of the case a = 0, without topological constraint, the surface undergoes a pinch-off and the final steady state is given by two spheres of radius ≈ 1 6 .…”

Section: Finite-element Simulationsmentioning

confidence: 99%

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Keeping it together: A phase-field version of path-connectedness and its implementation

Dondl

Wojtowytsch

2021

Journal of Algorithms & Computational Technology

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We describe the implementation of a topological constraint in finite-element simulations of phase-field models, which ensures path-connectedness of preimages of intervals in the phase-field variable. The constraint takes the form of an energetic penalty for a suitable geodesic distance between all pairs of points in the domain. The main application of our method presented here is a discrete steepest descent of a phase-field version of a bending energy with spontaneous curvature and additional surface area penalty. This leads to disconnected surfaces without our topological constraint but connected surfaces with the constraint. Numerically, our constraint is treated by first transforming the double integral over all pairs of points in the domain to a weighted graph structure and then using Dijkstra’s algorithm to calculate the distance between discrete connected components.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Finite-element Simulationsmentioning

confidence: 99%

Keeping it together: A phase-field version of path-connectedness and its implementation

Dondl

Wojtowytsch

2021

Journal of Algorithms & Computational Technology

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“…That is, the minimizers of and and the saddle point of are identical. In it was noted that for domains in

R^{2}

, which are smooth with the exception of convex corners, the weak solution and the Monk solution, as a variant of Ciarlet‐Raviart, coincide but that such coincidence is not guaranteed when the domain locally resembles a non‐convex sector. One aim of this note is to consider that situation.…”

Section: Comparison Of the Approachesmentioning

confidence: 99%

“…Already Babuška in was aware of possible issues for domains with corners, although the focus here lies mainly on the Navier boundary condition case. A survey of the problems that appear for the biharmonic differential equation on domains with corners can be found in .…”

Section: Introductionmentioning

confidence: 99%

Solving the biharmonic Dirichlet problem on domains with corners

Coster

Nicaise

Sweers

2014

Mathematische Nachrichten

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The biharmonic Dirichlet boundary value problem on a bounded domain is the focus of the present paper. By Riesz' representation theorem the existence and uniqueness of a weak solution is quite direct. The problem that we are interested in appears when one is looking for constructive approximations of a solution. Numerical methods using for example finite elements, prefer systems of second equations to fourth order problems. Ciarlet and Raviart in [7] and Monk in [21] consider approaches through second order problems assuming that the domain is smooth. We will discuss what happens when the domain has corners. Moreover, we will suggest a setting, which is in some sense between Ciarlet-Raviart and Monk, that inherits the benefits of both settings and that will give the weak solution through a system type approach.

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“…The second type is the mixed finite-element method [14,30] which only requires C 0 -conforming elements to approximate the solution. However, for the simply supported boundary condition, some mixed finite-element methods may result in spurious solutions on non-convex domains [15,18,43]. Besides, the solution of the saddle point problems resulting from the use of mixed finite-element method is also more involved than that for a direct discretization of the fourth-order operator.…”

Section: Introductionmentioning

confidence: 99%

$$C^1$$-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

Shan

Zhang

2019

Commun. Appl. Math. Comput.

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This paper is devoted to Professor Benyu Guo's open question on the C 1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C 1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C 1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C 1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C 1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.

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Corners Give Problems When Decoupling Fourth Order Equations Into Second Order Systems

Cited by 19 publications

References 11 publications

Keeping it together: A phase-field version of path-connectedness and its implementation

Keeping it together: A phase-field version of path-connectedness and its implementation

Solving the biharmonic Dirichlet problem on domains with corners

$$C^1$$-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

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