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This paper is the second of two articles, in which we aim to prove the convergence of the Hiptmair--Xu (HX) preconditioner (originally proposed by [R. Hiptmair and J. Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483--2509]) for H(curl)-elliptic boundary value problems with jump coefficients. In this paper, based on the auxiliary results obtained in our first article [Q. Hu, SIAM J. Numer. Anal., 59 (2021), pp. 2500--2535], we establish two new regular decompositions for lowdimensional edge finite element functions in three dimensions, which, under suitable assumptions on the distribution of the coefficients, are stable up to a polylogarithmic factor of the meshwidth with respect to weighted norms defined by the coefficients. Using these regular decompositions, we analyze the convergence of the HX preconditioner for the case of strongly discontinuous coefficients. We show that the HX preconditioner is asymptotically optimal up to a polylogarithmic factor in the meshwidth and is not severely affected by large jumps of the coefficients across the interface between two neighboring subdomains. [ABSTRACT FROM AUTHOR]

The a posteriori error estimate and greedy algorithms play central roles in the reduced basis method (RBM). In [M. Yano, Comput. Methods Appl. Mech. Engrg., 287 (2015), pp. 290--309; ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 163--185; SIAM J. Sci. Comput., 40 (2018), pp. A388--A420], several versions of RBMs based on exact error certifications and greedy algorithms with spatio-parameter adaptivities are developed. In this paper, with the parametric symmetric coercive elliptic boundary value problem as an example of the primal-dual variational problems satisfying the strong duality, we develop primal-dual RBMs (PD-RBM) with robust true error certifications and discuss three versions of greedy algorithms to balance the finite element error, the exact RB error, and the adaptive mesh refinements. For a class of convex minimization variational problems which has corresponding dual problems satisfying the strong duality, the primal-dual gap between the primal and dual functionals can be used as an a posteriori error estimator. This primal-dual gap error estimator is robust with respect to the parameters of the problem, and it can be used for both mesh refinements of finite element methods and the true RB error certification. With the help of an integration by parts formula, the primal-dual variational theory is developed for the symmetric coercive elliptic boundary value problems with nonhomogeneous boundary conditions by both the conjugate function and Lagrangian theories. A generalized Prager--Synge identity, which is the primal-dual gap error representation for this specific problem, is developed. RBMs for both the primal and dual problems with robust error estimates are developed. The dual variational problem often can be viewed as a constraint optimization problem. In the paper, different from the standard saddlepoint finite element approximation, the dual RBM is treated as a Galerkin projection by constructing RB spaces satisfying the homogeneous constraint. Inspired by the greedy algorithm with spatioparameter adaptivity of [M. Yano, SIAM J. Sci. Comput., 40 (2018), pp. A388--A420], adaptive balanced greedy algorithms with primal-dual finite element and RB error estimators are discussed. Numerical tests are presented to test the PD-RBM with adaptive balanced greedy algorithms. [ABSTRACT FROM AUTHOR]

In this paper we provide a generalization of the Douglas--Rachford splitting (DRS) and the primal-dual algorithm [L. Condat, J. Optim. Theory Appl., 158 (2013), pp. 460-479; B. C. V\~u, Adv. Comput. Math., 38 (2013), pp. 667-681] for solving monotone inclusions in a real Hilbert space involving a general linear operator. The proposed method allows for primal and dual nonstandard metrics and activates the linear operator separately from the monotone operators appearing in the inclusion. In the simplest case when the linear operator has full range, it reduces to classical DRS. Moreover, the weak convergence of primal-dual sequences to a Kuhn--Tucker point is guaranteed, generalizing the main result in [B. F. Svaiter, SIAM J. Control Optim., 49 (2011), pp. 280-287]. Inspired by [D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, M. Fortin and R. Glowinski, eds., Stud. Math. Appl. 15, North-Holland, Amsterdam, 1983, pp. 299-331], we also derive a new split alternating direction method of multipliers (SADMM) by applying our method to the dual of a convex optimization problem involving a linear operator which can be expressed as the composition of two linear operators. The proposed SADMM activates one linear operator implicitly and the other one explicitly, and we recover ADMM when the latter is set as the identity. Connections and comparisons of our theoretical results with respect to the literature are provided for the main algorithm and SADMM. The flexibility and efficiency of both methods is illustrated via numerical simulations in total variation image restoration and a sparse minimization problem. [ABSTRACT FROM AUTHOR]

This work is devoted to show the efficiency of a new numerical approach in solving geometrical shape optimization problems constrained to partial differential equations, on a family of convex domains. More precisely, we are interested to an improved numerical optimization process based on the new shape derivative formula, using the Minkowski deformation of convex domains, recently established in Boulkhemair and Chakib (J. Convex Anal. 21( n ∘ 1 ), 67–87 2014), Boulkhemair (SIAM J. Control Optim. 55( n ∘ 1 ), 156–171 2017). This last formula allows to express the shape derivative by means of the support function, in contrast to the classical one expressed in term of vector fields Henrot and Pierre 2005, Delfour and Zolésio 2011, Sokolowski and Zolesio 1992. This avoids some of the disadvantages related to the classical shape derivative approach, when one use the finite elements discretization for approximating the auxiliary boundary value problems in shape optimization processes Allaire 2007. So, we investigate here the performance of the proposed shape optimization approach through the numerical resolution of some shape optimization problems constrained to boundary value problems governed by Laplace or Stokes operator. Notably, we carry out a comparative numerical study between its resulting numerical optimization process and the classical one. Finally, we give some numerical results showing the efficiency of the proposed approach and its ability in producing good quality solutions and in providing better accuracy for the optimal solution in less CPU time compared to the classical approach. [ABSTRACT FROM AUTHOR]

This work studies O3 pollution for Chon Buri city in the eastern region of Thailand, where O3 has become an increased and serious concern in the last decade. It includes emission estimation and photochemical box modeling in support of investigating the underlying nature of O3 formation over the city and the roles of precursors emitted from sources. The year 2006 was considered and two single-day episodes (January 29 and February 14) were chosen for simulations. It was found that, in the city, the industrial sector is the largest emissions contributor for every O3 precursor (i.e., NOx, non-methane volatile organic compounds or NMVOC, and CO), followed by on-road mobile group. Fugitive NMVOC is relatively large, emitted mainly from oil refineries and tank farms. Simulated results acceptably agree with observations for daily maximum O3 level in both episodes and evidently indicate the VOC-sensitive regime for O3 formation. This regime is also substantiated by morning NMVOC/NOx ratios observed in the city. The corresponding O3 isopleth diagrams suggest NMVOC control alone to lower elevated O3. In seeking a potential O3 control strategy for the city, a combination of brute-force sensitivity tests, an experimental design, statistical modeling, and cost optimization was employed. A number of emission subgroups were found to significantly contribute to O3 formation, based on the February 14 episode, for example, oil refinery (fugitive), tank farm (fugitive), passenger car (gasoline), and motorcycle (gasoline). But the cost-effective strategy suggests control only on the first two subgroups to meet the standard. The cost of implementing the strategy was estimated and found to be small (only 0.2%) compared to the gross provincial product generated by the entire province where the city is located within. These findings could be useful as a needed guideline to support O3 management for the city. Supplemental Materials: Supplemental materials are available for this paper. Go to the publisher's online edition of the Journal of Air & Waste Management for a list of acronyms, outline of emission estimation, temporal allocation profiles, OZIPR initial and boundary conditions, and comparison of observations and simulated results. [ABSTRACT FROM AUTHOR]

This paper analyzes a class of singular control problems for which value functions are not necessarily smooth. Necessary and sufficient conditions for the well-known smooth fit principle, along with the regularity of the value functions, are given. Explicit solutions for the optimal policy and for the value functions are provided. In particular, when payoff functions satisfy the usual Inada conditions, the boundaries between action and continuation regions are smooth and strictly monotonic, as postulated and exploited in the existing literature (see [A. K. Dixit and R. S. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, NJ, 1994]; [M. H. A. Davis et al., Adv. in Appl. Probab., 19 (1987), pp. 156-176]; [T. Ø. Kobila, Stochastics Stochastics Rep., 43 (1993), pp. 29-63]; [A. B. Abel and J. C. Eberly, J. Econom. Dynam. Control, 21 (1997), pp. 831-852]; [A. Øksendal, Finance Stoch., 4 (2000), pp. 223-250]; [J. A. Scheinkman and T. Zariphopoulou, J. Econom. Theory, 96 (2001), pp. 180-207]; [A. Merhi and M. Zervos, SIAM J. Control Optim., 46 (2007), pp. 839-876]; and [L. H. Alvarez, A General Theory of Optimal Capacity Accumulation under Price Uncertainty and Costly Reversibility, Working Paper, Helsinki Center of Economic Research, Helsinski, Finland, 2006]). Illustrative examples for both smooth and nonsmooth cases are discussed to emphasize the pitfall of solving singular control problems with a priori smoothness assumptions. [ABSTRACT FROM AUTHOR]

The Regional Integrated Multi-Hazard Early Warning System (RIMES), an international, intergovernmental organization based in Thailand is engaged in disaster risk reduction over the Asia-Pacific region through early warning information. In this paper, RIMES' customized Weather Research Forecast (WRF) model has been used to evaluate the simulations of cyclone Nargis which hit Myanmar on 2 May 2008, the most deadly severe weather event in the history of Myanmar. The model covers a domain of 35ºE to 145ºE in the east-west direction and 12ºS to 40ºN in the north-south direction in order to cover Asia and east Africa with a resolution of 9 km in the horizontal and 28 vertical levels. The initial and boundary conditions for the simulations were provided by the National Center for Environmental Prediction-Global Forecast System (NCEP-GFS) available at 1º lon/lat resolution. An attempt is being made to critically evaluate the simulation of cyclone Nargis by seven set of simulations in terms of track, intensity and landfall time of the cyclone. The seven sets of model simulations were initialized every 12 h starting from 0000 UTC 28 April to 01 May 2008. Tropical Rainfall Measurement Mission (TRMM) precipitation (mm) is used to evaluate the performance of the simulations of heavy rainfall associated with the tropical cyclone. The track and intensity of the simulated cyclone are compared by making use of Joint Typhoon Warning Center (JTWC) data sets. The results indicate that the landfall time, the distribution and intensity of the rainfall, pressure and wind field are well simulated as compared with the JTWC estimates. The average landfall track error for all seven simulations was 64 km with an average time error of about 5 h. The average intensity error of central pressure in all the simulations were found out to be approximately 6 hPa more than the JTWC estimates and in the case of wind, the simulations under predicted it by an average of 12 m s. [ABSTRACT FROM AUTHOR]

Abstract: This paper establishes exponential decay bounds for steady magnetohydrodynamic geophysical finite pipe flow when homogeneous lateral surface boundary conditions are applied. In the spirit of earlier work of C.O. Horgan, L.T. Wheeler [Spatial decay estimates for the Navier–Stokes equations with application to the problem of entry flow, SIAM J. Appl. Math. 35 (1978) 97–116] and K.A. Ames, L.E. Payne [Decay estimates in steady pipe flow, SIAM J. Math. Anal. 20 (1989) 789–815] the decay to fully developed flow as a function of distance from the entry section is investigated. Here, it is not assumed that the flow is symmetric like in the work of J.C. Song, [Decay estimates for steady magnetohydrodynamic pipe flow, Nonlinear Anal. 54 (2003) 1029–1044] and it is fully developed at the exit section. Using a coupling parameter, we derive an integro-differential inequality that leads to estimates for the “energy” associated with the velocity and magnetic field represented by the difference between the entrance flow and the fully developed flow in a portion of the pipe near the exit section. This paper also indicates how to bound the total energy. [Copyright &y& Elsevier]

This article aims to finalize the classification of weakly well-posed hyperbolic boundary value problems in the half-space. Such problems with loss of derivatives are rather classical in the literature and appear for example in (Arch. Rational Mech. Anal.101 (1988) 261–292) or (In Analyse Mathématique et Applications (1988) 319–356 Gauthier-Villars). It is known that depending on the kind of the area of the boundary of the frequency space on which the uniform Kreiss–Lopatinskii condition degenerates then the energy estimate can include different losses. The three first possible areas of degeneracy have been studied in (Annales de l'Institut Fourier60 (2010) 2183–2233) and (Differential Integral Equations27 (2014) 531–562) by the use of geometric optics expansions. In this article we use the same kind of tools in order to deal with the last remaining case, namely a degeneracy in the glancing area. In comparison to the first cases studied we will see that the equation giving the amplitude of the leading order term in the expansion, and thus initializing the whole construction of the expansion, is not a transport equation anymore but it is given by some Fourier multiplier. This multiplier needs to be invert in order to recover the first amplitude. As an application we discuss the existing estimates of (Discrete Contin. Dyn. Syst., Ser. B23 (2018) 1347–1361; SIAM J. Math. Anal.44 (2012) 1925–1949) for the wave equation with Neumann boundary condition. [ABSTRACT FROM AUTHOR]

We analyze the convergence of piecewise collocation methods for computing periodic solutions of general retarded functional differential equations under the abstract framework recently developed in [S. Maset, Numer. Math., 133 (2016), pp. 525-555], [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2771-2793], and [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2794-2821]. We rigorously show that a reformulation as a boundary value problem requires a proper infinite-dimensional boundary periodic condition in order to be amenable to such analysis. In this regard, we also highlight the role of the period acting as an unknown parameter, which is critical since it is directly linked to the course of time. Finally, we prove that the finite element method is convergent, while we limit ourselves to commenting on the infeasibility of this approach as far as the spectral element method is concerned. [ABSTRACT FROM AUTHOR]

The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (Kim et al. in Peking Math J, 2020). Our methods consist of the generalization of the wellposedness theory for the Fokker–Planck equation (Hwang et al. SIAM J Math Anal 50(2):2194–2232, 2018; Hwang et al. Arch Ration Mech Anal 214(1):183–233, 2014) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi–Nash–Moser theory for the kinetic Fokker–Planck equations (Golse et al. Ann Sc Norm Super Pisa Cl Sci 19(1):253–295, 2019) and the Morrey estimates (Bramanti et al. J Math Anal Appl 200(2):332–354, 1996) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem. [ABSTRACT FROM AUTHOR]

We propose a new algorithm to improve the accuracy of spectral methods for singularly perturbed two-point boundary value problems. Driscoll and Hale [J. Numer. Anal., 36 (2016), pp. 108-132] suggest resampling as an alternative to row replacement when including boundary conditions. Testing this with an iterated sine-transformation [T. Tang and M. R. Trummer [SIAM J. Sci. Comput., 17 (1996), pp. 430-438] designed for boundary layers reveals artificial boundary conditions imposed by the transformation. The transformation is regularized to prevent this. The new regularized sine-transformation is employed to solve boundary value problems with and without resampling. It shows superior accuracy provided the regularization parameter is chosen from an optimal range. [ABSTRACT FROM AUTHOR]

We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirichlet (fractional and classical, respectively) and Neumann (fractional) boundary conditions. In particular, we develop a spectral penalty method (SPM) by using the Jacobi polyfractonomial approximation for the conservative R-L FDEs while using the polynomial approximation for the conservative Caputo FDEs. We establish the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems. This analysis allows us to estimate the proper values of the penalty parameters at boundary points. We present several numerical examples to verify the theory and demonstrate the high accuracy of the SPM, both for stationary and time dependent FDEs. Moreover, we compare the results against a Petrov-Galerkin spectral tau method (an extension of [Mao and Karniadakis, SIAM J. Numer. Anal., 56 (2018), pp. 24-49] and demonstrate the superior accuracy of SPM for all cases considered. [ABSTRACT FROM AUTHOR]