. Let an open bounded Lipschitz polygon or polyhedron Q, a function v in the Sobolev space H(div, Q), and a simplicial mesh of Q be given. We prove the equivalence of two piecewise (Raviart-Thomas) polynomial best approximations of v in the L-2-norm: (1) globally on the whole computational domain Q, with the normal trace continuity requirement and a divergence constraint; (2) locally on each mesh element, without any interelement continuity requirement and without any constraint on the divergence. The former (global-best continuous constrained piecewise polynomial approximation) arises in numerical methods for partial differential equations related to the H(div, Q) space, whereas the latter (local-best discontinuous unconstrained piecewise polynomial approximation) is a key quantity in approximation theory. Crucially, we establish p-robustness in that the equivalence constant only depends on the mesh shape regularity and the spatial dimension. This improves the recent result of Ern, Gudi, Smears, and Vohral & imath;k [IMA J. Numer. Anal. 42 (2022), pp. 1023-1049], where the equivalence constant was possibly dependent on the underlying polynomial degree. Consequently, we obtain fully h- and p- (mesh-size- and polynomial-degree-) optimal approximation estimates under the minimal Sobolev regularity only requested separately on each mesh element. These two results immediately follow by our construction of an operator from the infinite-dimensional Sobolev space H(div, Q) to its finite-dimensional Raviart-Thomas subspace that has the following properties: (1) it is defined over the entire H(div, Q) and preserves boundary conditions imposed on a part of the boundary of Q; (2) it is defined locally in a neighborhood of each mesh element; (3) it is based on elementwise L(2)orthogonal polynomial projections; (4) it is a projector, i.e., it leaves intact objects that are already in the Raviart-Thomas piecewise polynomial space; (5) it is locally and p-robustly stable in the L-2-norm, up to hp data oscillation; (6) its approximation property is locally and p-robustly equivalent to that of it satisfies the commuting property with the L-2-orthogonal projection onto piecewise polynomials.
This paper presents a rigorous convergence analysis of a stable numerical method for the three dimensional barotropic compressible Navier-Stokes (NS) system. In our approach, the discretization of the convection term is obtained using the standard upwind finite volume method. The continuity and momentum equations are approximated by the piecewise constant discontinuous Galerkin scheme and the H-1-conforming Bernardi-Raugel finite element method, respectively. There are two main contributions in this paper. Firstly, we rigorously prove that the present scheme converges to a weak solution of the barotropic NS equations; specifically, it includes the isentropic case (with the equation of state p(rho) = A rho(gamma)) which is one of the most important models in computational fluid dynamics (CFD). To achieve this, we derive enhanced consistency error bounds for the remainder terms through a refined counterattack technique coupled with sophisticated and intricate estimates. Furthermore, we develop a modified version of the discrete Lions-Feireisl existence theory (originally introduced by Karper). To the best of our knowledge, this work presents the first result that establishes convergence to weak solutions of the isentropic NS equations without requiring the constraint p'(0) > 0 or the pressure perturbation term. Secondly, we extend the analysis of the Crouzeix-Raviart finite element-discontinuous Galerkin (CR-DG) scheme to other numerical methods. The CR-DG scheme is the only one in the existing literature that has been proven to converge to the weak solution for the compressible NS equations, and extending its convergence analysis to other schemes has been mentioned as an open problem in several papers. The main difficulty is that the proof used in this research field relies heavily on the special structure of CR finite elements. We have overcome these structural limitations and demonstrated that our methodology can indeed be generalized to other possible schemes. Finally, some numerical experiments are conducted to verify the theoretical results.
We devise guidelines for deriving a posteriori error estimators for quasi-optimal nonconforming finite element methods and apply them in the case of symmetric elliptic problems of second order. The resulting estimators are defined for all source terms that are admissible to the underlying weak formulations. More importantly, they are equivalent to the error in a strict sense. In particular, their data oscillation part is bounded by the error and, furthermore, can be designed to be bounded by classical data oscillations. The estimators are computable, except for the data oscillation part. Since even the computation of some bound of the oscillation part is not possible in general, we advocate handling it on a case-by-case basis. We illustrate the practical use of two estimators obtained for the Crouzeix-Raviart method applied to the Poisson problem with a source term that is not a function and whose singular part with respect to the Lebesgue measure is not aligned with the mesh.
Recently, Giles et al. [Discrete Contin. Dyn. Syst. Ser. B 24 (2019), pp. 3881–3903] proved that the efficiency of the Multilevel Monte Carlo (MLMC) method for evaluating Down-and-Out barrier options for a diffusion process ( X t ) t ∈ [ 0 , T ] (X_t)_{t\in [0,T]} with globally Lipschitz coefficients can be improved by combining a Brownian bridge technique and a conditional Monte Carlo method provided that the running minimum inf t ∈ [ 0 , T ] X t \inf _{t\in [0,T]}X_t has a bounded density in the vicinity of the barrier. In the present work, thanks to the Lamperti transformation technique and using a Brownian interpolation of the drift implicit Euler scheme of Alfonsi [Probab. Lett. 83 (2013), pp. 602–607], we show that the efficiency of the MLMC can be also improved for the evaluation of barrier options for models with non-Lipschitz diffusion coefficients under certain moment constraints. We study two example models: the Cox-Ingersoll-Ross (CIR) and the Constant Elasticity of Variance (CEV) processes for which we show that the conditions of our theoretical framework are satisfied under certain restrictions on the models parameters. In particular, we develop semi-explicit formulas for the densities of the running minimum and running maximum of both CIR and CEV processes which are of independent interest. Finally, numerical tests are processed to illustrate our results.
We study the surface equations relating the sides, face diagonals and long diagonal of a rectangular box. It is an open problem whether a 'rational cuboid' exists, i.e., a rectangular box all of whose sides, face diagonals and long diagonal have (positive) rational length. The question is equivalent to the existence of non-trivial rational points on S & strns;. The surface S parametrizing cuboids: it is defined by the S has 48 isolated singular points. Let S be the minimal desingularization of S. The main result of this paper is the explicit determination of the Picard group of S, including its structure as a Galois module over Q. The main ingredient for showing that the known subgroup is actually the full Picard group is the use of the combined action of the Galois group and the geometric automorphism group of S (which we also determine) on the Picard group. This reduces the proof to checking that the hyperplane section is not divisible by 2 in the Picard group. We use our explicit knowledge of the Picard group, together with that of a K3 surface obtained as a quotient of S, to study curves of low degree on S & strns;. In this way, we completely classify all integral curves of degree at most 6 on S & strns;.
