A mixed finite element method for a sixth-order elliptic problem

Droniou, Jérôme; Ilyas, Muhammad; Lamichhane, Bishnu P.; Wheeler, Glen

doi:10.1093/imanum/drx066

Cited by 12 publications

(8 citation statements)

References 34 publications

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“…Theorem 2.1. For every φ h ∈ P 1 h (Ω) there exists a unique global minimum Φ ∈ H k (Ω) of the functional J defined in (2).…”

Section: Analytical Problem Formulationmentioning

confidence: 99%

“…In order to solve this numerically, we will reformulate this PDE of 6th order into a system of three PDEs of 2nd order. This is one of the possible techniques to handle higher order PDEs as demonstrated in [2] and [14]. This gives us the benefit of directly receiving the Laplacian as part of the solution.…”

Section: Reformulation From Higher Order To a System Of 2nd Ordermentioning

confidence: 99%

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A Tikhonov approach to level set curvature computation

Zvegincev¹

2022

Preprint

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In numerical simulations of two-phase flows, the computation of the curvature of the interface is a crucial ingredient. Using a finite element and level set discretization, the discrete interface is typically the level set of a low order polynomial, which often results in a poor approximation of the interface curvature. We present an approach to curvature computation using an approximate inversion of the L 2 projection operator from the Sobolev space H 2 or H 3 . For finite element computation of the approximate inverse, the resulting higher order equation is reformulated as a system of second order equations. Due to the Tikhonov regularization, the method is demonstrated to be stable against discretization irregularities. Numerical examples are shown for interior interfaces as well as interfaces intersecting the boundary of the domain.

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“…Theorem 2.1. For every φ h ∈ P 1 h (Ω) there exists a unique global minimum Φ ∈ H k (Ω) of the functional J defined in (2).…”

Section: Analytical Problem Formulationmentioning

confidence: 99%

Section: Reformulation From Higher Order To a System Of 2nd Ordermentioning

confidence: 99%

A Tikhonov approach to level set curvature computation

Zvegincev¹

2022

Preprint

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“…And some H m -nonconforming elements of lower degree on triangular meshes for any m were studied in [17]. In addition to standard conforming and nonconforming finite element methods, a C 0 interior penalty method in [14] and a cubic H 3nonconforming macro-element method in [18] were developed for a sixth-order elliptic equation in two dimensions, and some mixed finite element methods were advanced in [12,13,20] for 2mth-order elliptic equations with m > n. When m ≤ n, We refer to [3,27,28,15] for H m -conforming finite elements, and [23,22] for H m -nonconforming finite elements.…”

Section: Introductionmentioning

confidence: 99%

Nonconforming Virtual Element Method for $2m$-th Order Partial Differential Equations in $\mathbb R^n$ with $m>n$

Huang¹

2019

Preprint

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The H m -nonconforming virtual elements of any order k on any shape of polytope in R n with constraints m > n and k ≥ m are constructed in a universal way. A generalized Green's identity for H m inner product m > n is derived, which is essential to devise the H m -nonconforming virtual elements. The dimension of the H m -nonconforming virtual elements can reduced by the Serendipity approach. By means of the local H m projection and a stabilization term using the boundary degrees of freedom, the H m -nonconforming virtual element methods are proposed to approximate solutions of the m-harmonic equation. The norm equivalence of the stabilization on the kernel of the local H m projection is proved by using the bubble function technique, the Poincaré inquality and the trace inequality, which implies the well-posedness of the virtual element methods. Finally, the optimal error estimates for the H mnonconforming virtual element methods are achieved from an estimate of the weak continuity and the error estimate of the canonical interpolation.

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“…Nevertheless, some finite element schemes are developed for arbitrary order of derivative (i.e. [13,14,15]). For meshless methods based on the shape functions, there is no problem for the mesh construction in the higher dimensional space.…”

Section: Introductionmentioning

confidence: 99%

Higher order nonlocal operator method

Ren¹,

Zhuang²,

Rabczuk³

2019

Preprint

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We extend the nonlocal operator method to higher order scheme by using a higher order Taylor series expansion of the unknown field. Such a higher order scheme improves the original nonlocal operator method proposed by the authors in [A nonlocal operator method for solving partial differential equations], which can only achieve one-order convergence. The higher order nonlocal operator method obtains all partial derivatives with specified maximal order simultaneously without resorting to shape functions. The functional based on the nonlocal operators converts the construction of residual and stiffness matrix into a series of matrix multiplication on the nonlocal operator matrix. Several numerical examples solved by strong form or weak form are presented to show the capabilities of this method.

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A mixed finite element method for a sixth-order elliptic problem

Cited by 12 publications

References 34 publications

A Tikhonov approach to level set curvature computation

A Tikhonov approach to level set curvature computation

Nonconforming Virtual Element Method for $2m$-th Order Partial Differential Equations in $\mathbb R^n$ with $m>n$

Higher order nonlocal operator method

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