Non-convex Domains Research Articles

A singular local minimizer for the volume- constrained minimal surface problem in a nonconvex domain

It has recently been established by Wang and Xia that local minimizers of perimeter within a ball subject to a volume constraint must be spherical caps or planes through the origin. This verifies a conjecture of the authors and is in contrast to the situation of area-minimizing surfaces with prescribed boundary where singularities can be present in high dimensions. This result lends support to the more general conjecture that volume-constrained minimizers in arbitrary convex sets may enjoy better regularity properties than their boundary-prescribed cousins. Here, we show the importance of the convexity condition by exhibiting a simple example, given by the Simons cone, of a singular volume-constrained locally area-minimizing surface within a nonconvex domain that is arbitrarily close to the unit ball.

On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains

There exists a growing literature on using the Fokas method (unified transform method) to solve Laplace and Helmholtz problems on convex polygonal domains. We show here that the convexity requirement can be eliminated by the use of a ‘virtual side’ concept, thereby significantly increasing the flexibility and utility of the approach. We also show that the inclusion of singular functions in the basis to treat corner singularities can greatly increase the rate of convergence of the method. The method also compares well with other standard methods used to cope with corner singularities. An example is given where this inclusion leads to exponential convergence. As well as this, we give new results on several additional issues, including the choice of collocation points and calculation of solutions throughout domain interiors. An appendix illustrates the algebraic simplicity of the methodology by showing how the core part of the present approach can be implemented in only about a dozen lines of MATLAB code.

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

We study a new class of finite elements so‐called composite finite elements (CFEs), introduced earlier by Hackbusch and Sauter, Numer. Math., 1997; 75:447‐472, for the approximation of nonlinear parabolic equation in a nonconvex polygonal domain. A two‐scale CFE discretization is used for the space discretizations, where the coarse‐scale grid discretized the domain at an appropriate distance from the boundary and the fine‐scale grid is used to resolve the boundary. A continuous, piecewise linear CFE space is employed for the spatially semidiscrete finite element approximation and the temporal discretizations is based on modified linearized backward Euler scheme. We derive almost optimal‐order convergence in space and optimal order in time for the CFE method in the L∞(L2) norm. Numerical experiment is carried out for an L‐shaped domain to illustrate our theoretical findings.

Convergence of the Allen–Cahn equation with a zero Neumann boundary condition on non-convex domains

We study a singular limit problem of the Allen-Cahn equation with the homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data. The main result is the convergence of the time parametrized family of the diffused surface energy to Brakke's mean curvature flow with a generalized right angle condition on the boundary of the domain.

A grain-scale model for high-cycle fatigue degradation in polycrystalline materials

A grain-scale three-dimensional model for the analysis of fatigue intergranular degradation in polycrystalline materials is presented. The material microstructure is explicitly represented through Voronoi tessellations, of either convex or non-convex domains, and the mechanics of individual grains is modelled using a boundary integral formulation. The intergranular interfaces degrade under the action of cyclic loads and their behaviour is represented employing a cohesive zone model embodying a local irreversible damage parameter that evolves according to high-cycle continuum damage laws. The model is based on the use of a damage decomposition into static and cyclic contributions, an envelope load representation and a cycle jump strategy. The consistence between the cyclic damage and the action of the external loads, which contribute to the damage due to the redistribution of intergranular tractions between subsequent cycle jumps, is assessed at each solution step, so to capture the onset of macro-failure when the external actions cannot be equilibrated anymore by the critically damaged interfaces. Several numerical tests are reported to illustrate the potential of the developed method, which may find application in multiscale modelling of fatigue material degradation as well as in the design of micro-electro-mechanical devices (MEMS).

On the First Eigenvalue of the Degenerate $$\varvec{p}$$ p -Laplace Operator in Non-convex Domains

In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate p-Laplace operator, $$p>2$$ , in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincare–Sobolev inequalities. On this base we obtain lower estimates of the first non-trivial eigenvalues for Ahlfors-type domains (i.e. quasidiscs). This class of domains includes some snowflake-type domains with fractal boundaries.

Topology of Liouville Foliations for Integrable Billiards in Non-Convex Domains

Plane billiards are studied in non-convex domains bounded by arcs of confocal quadrics and in domains bounded by segments of mutually perpendicular straight lines. The topology of isoenergetic surfaces of such billiards is studied by calculating rough Liouville equivalence invariants known as Fomenko molecules.

CFS-PML-DEC formulation in two-dimensional convex and non-convex domains

In this paper, the time domain Maxwell’s equations are solved using the discrete exterior calculus (DEC) formalism in the two-dimensional space. To truncate the computational domain, the complex frequency-shifted perfectly matched layer (CFS-PML) concept is applied to create a reflectionless artificial layer. The paper presents a new numerical procedure to easily implement the CFS-PML with curved inner boundary. In order to numerically realize the PML, in a simplicial mesh, this paper proposes to utilize the nearest neighbor algorithm to associate point sets to boundary points. The distance from points to the boundary curve defines the attenuation function inside the PML. The performance of the approach is assessed by measuring the reflection error for three numerical experiments.

Decay of the Boltzmann Equation with the Specular Boundary Condition in Non-convex Cylindrical Domains

A basic question about the existence and stability of the Boltzmann equation in general non-convex domain with the specular reflection boundary condition has been widely open. In this paper, we consider cylindrical domains whose cross sections are general non-convex analytic planar domain. We establish the global-wellposedness and asymptotic stability of the Boltzmann equation with the specular reflection boundary condition in such domains. Our method consists of sharp classification of billiard trajectories which bounce infinitely many times or hit the boundary tangentially at some moment, and a delicate construction of an $\e$-tubular neighborhood of such trajectories. Analyticity of the boundary is crucially used. Away from such $\e$-tubular neighborhood, we control the number of bounces of trajectories and its' distance from singular sets in a uniform fashion. The worst case, sticky grazing set, can be excluded by cutting off small portion of the temporal integration. Finally we apply a method of \cite{KimLee} by the authors and achieve a pointwise estimate of the Boltzmann solutions.

Weighted finite element method for the Stokes problem with corner singularity

In this paper we introduce the notion of R ν -generalized solution to the Stokes problem with singularity in a non-convex polygonal domain with one reentrant corner of 3 π 2 on its boundary. The weighted analogue of the Ladyzhenskaya–Babus̆ka–Brezzi condition is proved. A new weighted finite element method is constructed. Results of numerical experiments have shown the efficiency of the method. • An R ν -generalized solution to the Stokes problem with corner singularity is introduced. • The weighted analogue of the LBB condition is proved. • A new weighted finite element method is constructed. • An approximate solution converges to an exact one with O ( h ) rate in W 2 , v 1 ( Ω ) seminorm.

Super-convergence and post-processing for mixed finite element approximations of the wave equation

We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consider the subsequent time discretization by the Crank-Nicolson method and show that the analysis and the post-processing strategy can be generalized to the fully discrete schemes. Our proofs do not rely on duality arguments or inverse inequalities and the results therefore apply also for non-convex domains and non-uniform meshes.

The point collocation method with a local maximum entropy approach

Meshless methods have long been a topic of interest in computational modelling in solid mechanics and are broadly divided into weak and strong form-based approaches. The need for numerical integration in the former remains a challenge often met by using a background mesh or complex stabilised nodal approaches. It is only strong form-based point collocation methods (PCMs) which dispense with meshing and integration entirely, and for this reason PCMs remain of interest. In this paper, a new point collocation method is developed which is based on maximum entropy basis functions which bring benefits in terms of accuracy and efficiency. These basis functions possess non-negativity and a weak Kronecker delta property which decreases the errors on boundaries to improve overall accuracy of solutions. After a discussion of implementation issues in the new method, numerical examples are presented, including 1D and 2D problems with linear elasticity and Poisson PDEs, on both convex and non-convex domains to show the performance. Comparisons of convergence properties with respect to accuracy and computational cost (both CPU time and floating point operations) are made with an existing method, the reproducing kernel collocation method (RKCM), to show the effectiveness of the proposed method. In all examples, higher order convergence rates are obtained using the developed method with increasingly reduced computational effort for higher levels of accuracy due to the fundamental advantages.

A Least-Squares/Relaxation Method for the Numerical Solution of the Three-Dimensional Elliptic Monge\u2013Amp\xe8re Equation

In this article, we address the numerical solution of the Dirichlet problem for the three-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a least-squares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and non-convex domains.

A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations

This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Karman equations defined on a polygonal domain. A discrete inf–sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established, and this allows the proof of local existence and uniqueness of a discrete solution to the nonlinear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel C0-interior penalty method (IPDG). In contrast to the known C0-IPDG dueto Brenner et al., (2016, A C0 interior penalty method for a von Karman plate. Numer. Math., 1–30), the overall discrete formulation maintains symmetry of the trilinear form in the first two components—despite the general nonsymmetry of the discrete nonlinear problems. Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known C0-IPDG lead to complications with some nonresidual-type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a nonconvex domain.

Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains

• Establish and prove some new definitions and lemmas of fractional derivative space on convex domains. • An unstructured mesh finite element method is presented. • The stability and convergence of the method are discussed on different irregular convex domains. • Extend the theory and develop a computational model for the case of a multiply-connected domain. • Apply to solve the coupled 2D fractional Bloch–Torrey equation on human brain-like domains. Fractional differential equations are powerful tools to model the non-locality and spatial heterogeneity evident in many real-world problems. Although numerous numerical methods have been proposed, most of them are limited to regular domains and uniform meshes. For irregular convex domains, the treatment of the space fractional derivative becomes more challenging and the general methods are no longer feasible. In this work, we propose a novel numerical technique based on the Galerkin finite element method (FEM) with an unstructured mesh to deal with the space fractional derivative on arbitrarily shaped convex and non-convex domains, which is the most original and significant contribution of this paper. Moreover, we present a second order finite difference scheme for the temporal fractional derivative. In addition, the stability and convergence of the method are discussed and numerical examples on different irregular convex domains and non-convex domains illustrate the reliability of the method. We also extend the theory and develop a computational model for the case of a multiply-connected domain. Finally, to demonstrate the versatility and applicability of our method, we solve the coupled two-dimensional fractional Bloch–Torrey equation on a human brain-like domain and exhibit the effects of the time and space fractional indices on the behaviour of the transverse magnetization.

A generalised comparison principle for the Monge–Ampère equation and the pressure in 2D fluid flows

We extend the generalised comparison principle for the Monge–Ampère equation due to Rauch & Taylor (1977) [15] to nonconvex domains. From the generalised comparison principle, we deduce bounds (from above and below) on solutions to the Monge–Ampère equation with sign-changing right-hand side. As a consequence, if the right-hand side is nonpositive (and does not vanish almost everywhere), then the equation equipped with a constant boundary condition has no solutions. In particular, due to a connection between the two-dimensional Navier–Stokes equations and the Monge–Ampère equation, the pressure p in 2D Navier–Stokes equations on a bounded domain cannot satisfy Δp≤0 in Ω unless Δp≡0 (at any fixed time). As a result, at any time t>0 there exists z∈Ω such that Δp(z,t)=0.

Dual <inline-formula> <tex-math notation="LaTeX">${M}$ </tex-math> </inline-formula>-Convex Variable Subsets Family and Extremum Analysis for the OPF Problem

The optimal power flow (OPF) model is a central optimization problem in power system network. In this paper, we propose a novel approach to solve the OPF problem that has a convex objective function and non-convex feasible domain due to the constraints. Based on the concept of abstract convex analysis, we construct the dual ${M}$ -convex subsets family of original variables by using the variable separation method, followed by the analysis of the extremum according to the infimum base of the ${M}$ -convex subsets. It is challenging to obtain an explicit mapping function among the separated variables due to the nonlinear equality constraints. We, therefore, use the theorem of implicit function and the differential function to do duality analysis on separated variables. Based on the min–max principle of the primal-dual problem of OPF, we derive the condition of the complementary factor which leads the Lagrange dual problem to maximum, and make the variable separation that results in a Minkowski-type dual ${M}$ -convex subset. We then can obtain a local minimum using the principle of abstract convex optimization, which will be the global optimal solution under the Karush–Kuhn–Tucker conditions. We evaluate the proposed approach on several IEEE systems. The simulation results indicate that the approach is feasible and effective to deal with the non-convex OPF problem with a non-convex feasible domain.

Semidiscrete Finite Element Analysis of Time Fractional Parabolic Problems: A Unified Approach

In this paper, we consider the numerical approximation of time-fractional parabolic problems involving Caputo derivatives in time of order $\alpha$, $0< \alpha<1$. We derive optimal error estimates for semidiscrete Galerkin finite element (FE) type approximations for problems with smooth and nonsmooth initial data. Our analysis relies on energy arguments and exploits the properties of the inverse of the associated elliptic operator. We present the analysis in a general setting so that it is easily applicable to various spatial approximations such as conforming and nonconforming FE methods (FEMs) and FEM on nonconvex domains. The finite element approximation in mixed form is also presented and new error estimates are established for smooth and nonsmooth initial data. Finally, an extension of our analysis to a multiterm time-fractional model is discussed.

Chemotactic Aggregation versus Logistic Damping on Boundedness in the 3D Minimal Keller--Segel Model

We study chemotactic aggregation versus logistic damping on boundedness for the 3D minimal Keller--Segel (KS) model with logistic source, $\{ u_t = \nabla \cdot (\nabla u-\chi u\nabla v)+ u-\mu u^2, \, x\in \Omega, t>0; v_t =\Delta v -v+u, \, x\in \Omega, t>0\}, $ in a smooth, bounded, but not necessarily convex domain $\Omega\subset \mathbb{R}^3$ with $\chi, \mu>0$, nonnegative initial data $u_0, v_0$, and homogeneous Neumann boundary data. In a previous work [T. Xiang, J. Math. Anal. Appl., 459 (2018), pp. 1172--1200], the global boundedess of the above KS system in nonconvex domains is guaranteed under the explicit condition $ (*) \;\mu>\vartheta_0\chi=\frac{9}{\sqrt{10}-2}\chi. %\eqno(*) $ As a continuation, under the critical condition $(*)$, up to a scaling constant depending only on $u_0$, $v_0$, and $\Omega$, we here establish explicit uniform-in-time upper bounds for the quantities $\|u(\cdot,t)\|_{L^\infty(\Omega)}$ and $\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}$ in terms of $\chi$ and $\mu$; these bounds are defined for all $\chi\geq 0$ and $\mu>\vartheta_0\chi$, increasing in $\chi$ and decreasing in $\mu$, and have only one singular line in $\mu$ at $\mu=\vartheta_0\chi$. The corresponding 2D qualitative boundedness has been investigated [H. Jin and T. Xiang, C. R. Math. Acad. Sci. Paris, 356 (2018), pp. 875--885], wherein the only singular line of the upper bounds in $\mu$ is shown to be $\mu=0$. By comparison, an interesting new feature is that the singular line $\mu=0$ of the corresponding 2D upper bounds has moved up to $\mu=\vartheta_0\chi$ in the 3D setting. It is worth mentioning that, in the absence of a logistic source, the corresponding classical KS model (by setting $\mu=0$ and removing the proliferation term $+u$ in the first equation) is now well known to possess blow-up solutions for even small initial data [M. Winkler, J. Math. Pures Appl., 100 (2013), pp. 748--767].

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasi-uniform meshes and superconvergence meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in non-convex domains are provided.