A Taylor Series Condition for Harmonic Extension

Coffman, Adam; Legg, David A.; Pan, Yifei

doi:10.14321/realanalexch.28.1.0235

Cited by 7 publications

(8 citation statements)

References 9 publications

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“…Combining ( 12) and ( 13), provided h ≤ (2c) −1 we establish (9) with α = 1/3. Now let u H ∈ H 1 (Ω) be the solution of the Laplace equation ∆u H = 0 in Ω fulfilling u H = g on Γ.…”

Section: The Approximate Problemmentioning

confidence: 65%

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Methods of arbitrary optimal order with tetrahedral finite-element meshes forming polyhedral approximations of curved domains

Ruas

2017

Preprint

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In recent papers (see e.g. [22], [25], [26]) the author introduced a simple alternative to isoparametric finite elements of the n-simplex type, to enhance the accuracy of approximations of secondorder boundary value problems with Dirichlet conditions, posed in smooth curved domains. This technique is based upon trial-functions consisting of piecewise polynomials defined on straightedged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast the test-functions are defined upon the standard degrees of freedom associated with the underlying method for polytopic domains. While method's mathematical analysis for both second-and fourth-order problems in two-dimensional domains was carried out in [22] and [27], this paper is devoted to the study of the three-dimensional case, in which the method is nonconforming. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are demonstrated for a tetrahedron-based Lagrange family of finite elements. Unprecedented L 2 -error estimates for the class of problems considered in this work are also proved. A series of numerical examples illustrates the potential of the new technique. In particular its better accuracy at equivalent cost as compared to the isoparametric technique is highlighted. Moreover the great generality of the new approach is exemplified through a method with degrees of freedom other than nodal values.

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Section: The Approximate Problemmentioning

confidence: 65%

“…Akin to problem (8) and thanks to (9), problem (92) has a unique solution. This fact allows us to claim the following preliminary result:…”

Section: The Case Of Non-convex Domainsmentioning

confidence: 99%

Methods of arbitrary optimal order with tetrahedral finite-element meshes forming polyhedral approximations of curved domains

Ruas

2017

Preprint

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“…Even if we go the other around by prescribing a regular f in

\overset{Ω}{true}

, the existence of an associated

\overset{u}{true}

fulfilling the assumptions of Theorem 4.5 can be questioned. However using results in [9] combined with standard ones (cf. [13]) it is possible to identify cases where such an extension

\overset{u}{true}

does exist.…”

Section: Error Estimatesmentioning

confidence: 99%

Optimal Lagrange and Hermite finite elements for Dirichlet problems in curved domains with straight‐edged triangles

Ruas

2020

Z Angew Math Mech

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One of the reasons for the success of the finite element method in Solid Mechanics, among other Applied Sciences, is its versatility to deal with bodies of arbitrary shape. In case the problem at hand is modeled by second‐order partial differential equations with Dirichlet conditions prescribed on a curvilinear boundary, method's isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for polygonal or polyhedral domains and methods of order greater than one based on standard straight‐edged elements. However, besides algebraic and geometric inconveniences, the isoparametric technique is limited in scope, since its extension to degrees of freedom other than function values is not straightforward. In previous work the author introduced a simple alternative, which bypasses the above drawbacks without eroding qualitative approximation properties. Among other advantages, it can do without curved elements and is based only on polynomial Algebra. The purpose of this paper is two‐fold: On the one hand the mathematical foundations of this method are established, in the framework of a model Poisson problem solved with Lagrange elements; in particular optimal‐rate error estimates both in energy norm and in the mean‐square sense are demonstrated and a numerical comparison with the isoparametric technique is carried out. On the other hand this study is extended to Hermite elements with normal‐derivative degrees of freedom for solving the equation modeling the bending of clamped thin plates with a curvilinear edge.

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“…Let r 0 be the radius of the largest (open) ball B contained in Ω and O = (x 0 , y 0 ) be its center. Assuming that the extension of f is not too wild in Ω so that the Taylor series of u H (x, y 0 ) and [∂u H /∂y](x, y 0 ) centered at O converge in the segment of the line y = y 0 centered at O with length equal to r 0 √ 2r 0 + 2δ for a certain δ > 0, according to [9] there exists a harmonic extension u H of u H to the ball B 0 centered at O with radius r 0 + δ √ 2. Clearly in this case, as long as δ is large enough for B to contain Ω, we can define ũ := u 0 − u H as a function in H k+1 ( Ω) that vanishes on Γ, and thus satisfies all the required properties.…”

Section: Error Estimatesmentioning

confidence: 99%

Optimal simplex finite-element approximations of arbitrary order in curved domains circumventing the isoparametric technique

Ruas¹

2017

Preprint

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Since the 1960's the finite element method emerged as a powerful tool for the numerical simulation of countless physical phenomena or processes in applied sciences. One of the reasons for this undeniable success is the great versatility of the finite-element approach to deal with different types of geometries. This is particularly true of problems posed in curved domains of arbitrary shape. In the case of function-value Dirichlet conditions prescribed on curvilinear boundaries method's isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for standard straight elements in the case of polygonal or polyhedral domains. However, besides obvious algebraic and geometric inconveniences, the isoparametric technique is helplessly limited in scope and simplicity, since its extension to degrees of freedom other than function values is not straightforward if not unknown. The purpose of this paper is to propose, study and test a simple alternative that bypasses all the above drawbacks, without eroding qualitative approximation properties. More specifically this technique can do without curved elements and is based only on polynomial algebra.

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A Taylor Series Condition for Harmonic Extension

Cited by 7 publications

References 9 publications

Methods of arbitrary optimal order with tetrahedral finite-element meshes forming polyhedral approximations of curved domains

Methods of arbitrary optimal order with tetrahedral finite-element meshes forming polyhedral approximations of curved domains

Optimal Lagrange and Hermite finite elements for Dirichlet problems in curved domains with straight‐edged triangles

Optimal simplex finite-element approximations of arbitrary order in curved domains circumventing the isoparametric technique

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