Convex Polyhedral Domains Research Articles

Stable determination of an elastic medium scatterer by a single far-field measurement and beyond

We are concerned with the time-harmonic elastic scattering due to an inhomogeneous elastic material inclusion located inside a uniformly homogeneous isotropic medium. We establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in \({\mathbb {R}}^n\), \(n=2,3\). Our argument in establishing the stability result is localized around a corner of the medium scatterer. This enables us to further establish a byproduct result by proving that if a generic medium scatterer, not necessary to be a polyhedral shape, possesses a corner, then there exists a positive lower bound of the scattered far-field patterns. The latter result indicates that if an elastic material object possesses a corner on its support, then it scatters every incident wave stably and invisibility phenomenon does not occur.

On the analysis and approximation of some models of fluids over weighted spaces on convex polyhedra

We study the Stokes problem over convex polyhedral domains on weighted Sobolev spaces. The weight is assumed to belong to the Muckenhoupt class \(\varvec{A}_{\varvec{q}}\) for \(\varvec{q} \in (1,\varvec{\infty })\). We show that the Stokes problem is well-posed for all \(\varvec{q}\). In addition, we show that the finite element Stokes projection is stable on weighted spaces. With the aid of these tools, we provide well-posedness and approximation results to some classes of non-Newtonian fluids.

A Convergent Post-processed Discontinuous Galerkin Method for Incompressible Flow with Variable Density

We propose a linearized semi-implicit and decoupled finite element method for the incompressible Navier--Stokes equations with variable density. Our method is fully discrete and shown to be unconditionally stable. The velocity equation is solved by an H1-conforming finite element method, and an upwind discontinuous Galerkin finite element method with post-processed velocity is adopted for the density equation. The proposed method is proved to be convergent in approximating reasonably smooth solutions in three-dimensional convex polyhedral domains.

The optimal convergence of finite element methods for the three-dimensional second-order elliptic boundary value problem with singularity

Let x0∈Ω. Assume that Gx0(x) satisfies the elliptic boundary value problem LGx0(x)≡∂∂xj(aij(x)∂Gx0(x)∂xi)=δ(x−x0),inΩ,Gx0(x)=0on∂Ω, where Ω⊂ℜ3 is a convex polyhedral domain. In this article, we propose specially graded meshes Th. Furthermore, we obtain an L2(Ω)-error estimate of order hk+1|ln⁡h|32 for approximations with a Pk finite element method.

Pathwise differentiability of reflected diffusions in convex polyhedral domains

Les diffusions réfléchies dans les domaines polyhédriques convexes apparaissent dans diverses applications, notamment les systèmes de particules, les réseaux de files d’attente, les réseaux de réaction biochimique et la finance mathématique. Dans des conditions appropriées sur les données, nous établissons la différentiabilité selon la trajectoire d’une telle diffusion réfléchie par rapport à ses paramètres de définition, tel que sa condition initiale, ses coefficients de dérive et de diffusion et ses directions (obliques) de réflexion le long des limites du domaine. Nous caractérisons la régularisation continue droite d’une dérivée selon la trajectorie de la diffusion réfléchie comme solution unique à une équation différentielle stochastique linéaire contrainte avec sauts dont les coefficients de dérive et de diffusion, le domaine et les directions de réflexion dépendent de l’état de la diffusion réfléchie. Les travaux antérieurs dans le contexte multidimensionnel ont été en grande partie limités à l’étude de la différentiabilité des flux stochastiques des mouvements browniens (normalement) réfléchis. Une difficulté essentielle consiste à identifier une linéarisation appropriée de la dynamique du processus de temps local, en particulier en présence d’une frontière non lisse. Nous adoptons une nouvelle approche qui utilise les propriétés des dérivées directionnelles de l’application de Skorokhod étendue associée et leur caractérisation en termes d’une problème dérivé. La preuve implique l’établissement de certaines propriétés des trajectoires de la diffusion réfléchie aux frontieres non lisses du domaine polyhédral, qui peut être un resultat d’intérêt indépendant, et de démontrer que les dérivées selon la trajectorie des diffusions réfléchies peuvent être caractérisées en termes de dérivées directionnelles de l’application de Skorokhod étendu. En corollaire, nous obtenons une représentation probabiliste pour les dérivées des valeurs d’attendues des fonctionnelles de diffusions réfléchies, ce qui est utile pour l’analyse de sensibilité des diffusions réfléchies.

A Monte Carlo Method for Estimating Sensitivities of Reflected Diffusions in Convex Polyhedral Domains

We develop an effective Monte Carlo method for estimating sensitivities or gradients of expectations of sufficiently smooth functionals of a reflected diffusion in a convex polyhedral domain with respect to its defining parameters, namely its initial condition, drift, and diffusion coefficients and directions of reflection. Our method, which falls into the class of infinitesimal perturbation analysis (IPA) methods, uses a probabilistic representation for such sensitivities as the expectation of a functional of the reflected diffusion and its associated derivative process. The latter process is the unique solution to a constrained linear stochastic differential equation with jumps whose coefficients, domain, and directions of reflection are modulated by the reflected diffusion. We propose a consistent estimator for such sensitivities using an Euler approximation of the reflected diffusion and its associated derivative process. Proving that the Euler approximation converges is challenging because the derivative process jumps whenever the reflected diffusion hits the boundary of the domain. A key step in the proof is establishing a continuity property of the related derivative map, which is of independent interest. We compare the performance of our IPA estimator with a standard likelihood ratio estimator (whenever the latter is applicable) and provide numerical evidence that the variance of the former is substantially smaller than that of the latter. We illustrate our method with an example of a rank-based interacting diffusion model of equity markets. Interestingly, we show that estimating certain sensitivities of the rank-based interacting diffusion model using our method for a reflected Brownian motion description of the model outperforms a finite difference method for a stochastic differential equation description of the model.

A new analysis of discontinuous Galerkin methods for a fourth order variational inequality

We study a family of discontinuous Galerkin methods for the displacement obstacle problem of Kirchhoff plates on two and three dimensional convex polyhedral domains, which are characterized as fourth order elliptic variational inequalities of the first kind. We prove that the error in an H2-like energy norm is O(hα) for the quadratic method, where α∈(12,1] is determined by the geometry of the domain. Under additional assumptions on the contact set such that the solution has improved regularity, we derive the optimal error estimate with α∈(1,32) for the cubic method. Numerical experiments demonstrate the performance of the methods and confirm the theoretical results.

Examples of tropical-to-Lagrangian correspondence

The paper associates Lagrangian submanifolds in symplectic toric varieties to certain tropical curves inside the convex polyhedral domains of $$\mathbb {R}^n$$ that appear as the images of the moment map of the toric varieties. We pay a particular attention to the case $$n=2$$ , where we reprove Givental’s theorem (Givental in Funct Anal Appl 20(3):197–203, 1986) on Lagrangian embeddability of non-oriented surfaces to $$\mathbb {C}^2$$ , as well as to the case $$n=3$$ , where we see appearance of the graph 3-manifolds studied by Waldhausen (I Invent Math 3:308–333, 1967a, II Invent Math 4:87–117, 1967b) as Lagrangian submanifolds. In particular, rational tropical curves in $$\mathbb {R}^3$$ produce 3-dimensional rational homology spheres. The order of their first homology groups is determined by the multiplicity of tropical curves in the corresponding enumerative problems.

On directional derivatives of Skorokhod maps in convex polyhedral domains

The study of both sensitivity analysis and differentiability of the stochastic flow of a reflected process in a convex polyhedral domain is challenging due to the abrupt change in the nature of the dynamics at the boundary and is further complicated because the boundary is not smooth. These difficulties can be addressed by studying directional derivatives of an associated extended Skorokhod map, which is a deterministic mapping that takes an unconstrained path to a suitably reflected or constrained version. In this work, we develop an axiomatic framework for the analysis of directional derivatives of a large class of Lipschitz continuous extended Skorokhod maps in convex polyhedral domains with oblique directions of reflection. We establish existence of directional derivatives at a path whose reflected version satisfies a certain boundary jitter property, and also show that the right-continuous regularization of such a directional derivative can be characterized as the unique solution to a Skorokhod-type problem, where both the domain and directions of reflection vary (discontinuously) depending on the state of the reflected path. A key step in the analysis is the proof of certain contraction properties for a family of (oblique) derivative projection operators. The results of this paper are used in subsequent work to study differentiability of stochastic flows and sensitivity analysis for a large class of reflected diffusions in convex polyhedral domains.

Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra

In general polygons and polyhedra, possibly nonconvex, the analyticity of the finite element heat semigroup in the $L^q$ norm, $1\leq q\leq\infty$, and the maximal $L^p$-regularity of semi-discrete finite element solutions of parabolic equations are proved. By using these results, the problem of maximum-norm stability of the finite element parabolic projection is reduced to the maximum-norm stability of the Ritz projection, which currently is known to hold for general polygonal domains and convex polyhedral domains.

A Semidiscrete Finite Element Approximation of a Time-Fractional Fokker--Planck Equation with NonSmooth Initial Data

We present a new stability and convergence analysis for the spatial discretization of a time-fractional Fokker--Planck equation in a convex polyhedral domain, using continuous, piecewise-linear, finite elements. The forcing may depend on time as well as on the spatial variables, and the initial data may have low regularity. Our analysis uses a novel sequence of energy arguments in combination with a generalized Gronwall inequality. Although this theory covers only the spatial discretization, we present numerical experiments with a fully discrete scheme employing a very small time step, and observe results consistent with the predicted convergence behavior.

Symmetric interior penalty Galerkin approaches for two-dimensional parabolic interface problems with low regularity solutions

This work presents novel finite element approaches for solving a parabolic partial differential equation with discontinuous coefficients and low regularity solutions in a bounded convex polyhedral domain. A spatial semi-discretization based on symmetric interior penalty Galerkin (SIPG) approximations is constructed and analyzed by using discontinuous piecewise linear functions. For smooth initial data, spatial errors in the broken L2, H1 and L2(H1) norms are proven to be optimal with respect to low regularity solutions, which are only piecewise H1+s smooth with 0<s≤1. Furthermore we present the full SIPG discretizations based on an Euler backward finite difference time discretization. The proposed approximations are shown to be unconditionally stable, and have nearly the optimal L2(L2) and L2(H1) error estimates, even when the regularities of the solutions are low on the whole domain. The error estimates are optimal with respect to time, sharply depending on the indexes in the global and local regularity. Numerical experiments for two-dimensional parabolic interface problems verify the theoretical convergence rates.

On unique determination of conformal type for domains in Euclidean spaces

This survey is devoted to discussion of problems of the unique determination of conformal type for domains in Euclidean spaces. It consists of three parts. In the first of them, we consider results on the problem of the unique determination for (generally speaking) nonconvex domains in R n , where n ≥ 4. The second part is devoted to study of problems on unique determination of convex polyhedral domains in three-dimensional Euclidean space by relative conformal moduli of boundary condensers. Finally, in the third part, we study problems of unique determination of 3-connected plane domains by relative conformal moduli of pairs of boundary components.

A New Convergence Analysis of Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints

We consider finite element methods for elliptic distributed optimal control problems with pointwise state constraints on two and three dimensional convex polyhedral domains formulated as fourth order variational inequalities. We develop a new convergence analysis that is applicable to $C^1$ finite element methods, classical nonconforming finite element methods and discontinuous Galerkin methods.

L p -Regularity for the Cauchy-Dirichlet Problem for Parabolic Equations in Convex Polyhedral Domains

In this paper, we study the regularity of the solution of the initial-boundary value problem with Dirichlet boundary conditions for second-order divergence parabolic equations in a domain of polyhedral type. We establish several results on the regularity of the solution in weighted Lp-Sobolev spaces.

Finite Element Pointwise Results on Convex Polyhedral Domains

The main goal of the paper is to establish that the $L^1$ norm of jumps of the normal derivative across element boundaries and the $L^1$ norm of the Laplacian of a piecewise polynomial finite element function can be controlled by corresponding weighted discrete $H^2$ norm on convex polyhedral domains. In the finite element literature such results are only available for piecewise linear elements in two dimensions and the extension to convex polyhedral domains is rather technical. As a consequence of this result, we establish almost pointwise stability of the Ritz projection and the discrete resolvent estimate in $L^\infty$ norm.

Quasi-optimality of adaptive finite element methods for controlling local energy errors

A rich theory demonstrating convergence and optimality of adaptive finite element methods (AFEM) has been developed in recent years. In this work we prove optimality of AFEM which are designed to control local energy errors in elliptic partial differential equations. Because errors propagate globally in FEM, controlling local errors requires controlling both local energy solution properties and global error contributions (pollution errors) which may be measured in a weaker norm such as the $$L_2$$L2 norm. We define adaptive methods which control both of these error components and prove that they converge with the best possible rate over all possible refinements of the initial mesh. These results are valid for Poisson's problem on convex polyhedral domains in arbitrary space dimension. Our theory establishes AFEM optimality for several adaptive marking strategies which rigorously control pollution effects. We also present numerical examples that illustrate our theory and confirm that local energy AFEM without pollution control can fail to yield optimal meshes.

The Galerkin finite element method for a multi-term time-fractional diffusion equation

We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one- and two-dimensional problems confirm the theoretical convergence rates.

Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion

We consider the initial boundary value problem for the inhomogeneous time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and a nonsmooth right hand side data in a bounded convex polyhedral domain. We analyze two semidiscrete schemes based on the standard Galerkin and lumped mass finite element methods. Almost optimal error estimates are obtained for right hand side data $f(x,t)\in L^\infty(0,T;\dot H^q(\Omega))$, $-1< q \le 1$, for both semidiscrete schemes. For lumped mass method, the optimal $L^2(\Omega)$-norm error estimate requires symmetric meshes. Finally, numerical experiments for one- and two-dimensional examples are presented to verify our theoretical results.

On uniform positivity of transition densities of small noise constrained diffusions

Constrained diffusions in convex polyhedral domains with a general oblique reflection field, and with a diffusion coefficient scaled by a small parameter, are considered. Using an interior Dirichlet heat kernel lower bound estimate for second order elliptic operators in bounded domains from [13], certain uniform in the scaling parameter lower bounds on transition densities of such constrained diffusions are established. These lower bounds together with results from [1] give, under additional stability conditions, an exponential leveling property, as the scaling parameter approaches zero, for exit times from suitable bounded domains.