A-priori analysis and the finite element method for a class of degenerate elliptic equations

Li, Hengguang

doi:10.1090/s0025-5718-08-02179-0

Cited by 20 publications

(8 citation statements)

References 41 publications

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“…Degenerate elliptic equations have many applications in Numerical Analysis, see [13,38,41,88,87], for example.…”

Section: 3mentioning

confidence: 99%

Analysis on singular spaces: Lie manifolds and operator algebras

Nistor

2016

Journal of Geometry and Physics

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We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds--called "Lie manifolds"--that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here--work that spans over close to two decades--was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.Comment: 43 pages, based on my four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 201

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“…Degenerate elliptic equations have many applications in Numerical Analysis, see [13,38,41,88,87], for example.…”

Section: 3mentioning

confidence: 99%

Analysis on singular spaces: Lie manifolds and operator algebras

Nistor

2016

Journal of Geometry and Physics

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“…Analysis on this type of manifolds is related to that on hyperbolic spaces [32]. It is also related to the analysis of edge singularities for PDEs on polyhedral domains [7,13,16,20,42].…”

Section: In View Of Equationsmentioning

confidence: 99%

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

2014

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We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a "small" parameter. As an example, we treat in detail the important case of the SABR PDE for β = 1, namely ∂τ u = σ 2 1 2 (∂ 2The terms u j are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of u that are in "closed form," and hence can be evaluated very quickly. Most of the other related methods use the "time" τ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of ν, similar to Hagan's formula, but including also the mean reverting term. We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility. Contents32 5.2. The SABR model (with general β) 34 References 35

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“…Nevertheless, techniques for the estimation of the finite element approximation for boundary value problems with singular coefficients can be found in the papers of Eriksson and Thomée [28], Franchi and Tesi [30], Li [39], and in the references therein. Also, Bespalov and Rukavishnikov [15,57] studied the p-version finite element approximation in the case when V has a single singularity at the origin, if the origin is a boundary point.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM

Nistor

2009

Journal of Computational and Applied Mathematics

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a b s t r a c tLet r = (x 2 1 + x 2 2 ) 1/2 be the distance function to the origin O ∈ R 2 , and let us fix δ > 0. We consider the ''Schrödinger-type mixed boundary value problem'' −∆u + δr −2 u = f ∈ H m−1 (Ω) on a bounded polygonal domain Ω ⊂ R 2 . The singularity in the potential δr −2 severely limits the regularity of the solution u. This affects the rate of convergence to u of the finite element approximations u S ∈ S obtained using a quasi-uniform sequence of meshes. We show that a suitable graded sequence of meshes recovers the quasi-optimal, where S n are the FE spaces of continuous, piecewise polynomial functions of degree m ≥ 1 associated to our sequence of meshes and u n = u Sn ∈ S n are the FE approximate solutions. This is in spite of the fact that u ∈ H m+1 (Ω) in general. One of the main results of our paper is to show that the singularities due to the potential and the singularities due to the singularities of the domain or to the change in boundary conditions can be treated in the same way. Our proof is based on regularity and well-posedness results in weighted Sobolev spaces, with the weight taking into account all singularities (including the ones coming from the potential). Our regularity results apply also to operators with weaker singularities, like the Schrödinger operator −∆u + δr −1 , for which we also obtain Fredholm conditions and a formula for the index. Our a priori estimates also extend to piecewise smooth domains (i.e., curvilinear polygonal domains).

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A-priori analysis and the finite element method for a class of degenerate elliptic equations

Cited by 20 publications

References 41 publications

Analysis on singular spaces: Lie manifolds and operator algebras

Analysis on singular spaces: Lie manifolds and operator algebras

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM

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