• Abstract

Motivated by the study of the corner singularities in the so-called cavity flow, we establish in the first part of this article, the existence and uniqueness of solutions in L²(Ω)² for the Stokes problem in a domain Ω, when Ω is a smooth domain or a convex polygon. This result is based on a new trace theorem and we show that the trace of u can be arbitrary in L²(∂Ω)² except for a standard compatibility condition recalled below. The results are also extended to the linear evolution Stokes problem. Then in the second part, using a finite element discretization, we present some numerical simulations of the Stokes equations in a square modeling thus the well known lid-driven flow. The numerical solution of the lid driven cavity flow is facilitated by a regularization of the boundary data, as in other related equations with corner singularities ([9], [10], [45], [24]). The regularization of the boundary data is justified by the trace theorem in the first part.

• Abstract

In this paper, we consider the discontinuous Galerkin (DG) methods to solve linear hyperbolic equations with singular initial data. With the help of weight functions, the superconvergence properties outside the pollution region will be investigated. We show that, by using piecewise polynomials of degree k and suitable initial discretizations, the DG solution is (2k+1)-th order accurate at the downwind points and (k+2)-th order accurate at all the other downwindbiased Radau points. Moreover, the derivative of error between the DG and exact solutions converges at a rate of k+1 at all the interior upwind-biased Radau points. Besides the above, the DG solution is also (k+2)-th order accurate towards a particular projection of the exact solution and the numerical cell averages are (2k+1)-th order accurate. Numerical experiments are presented to confirm the theoretical results.

• Abstract

In this paper, we provide the optimal convergence rate of a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the second-order wave equation in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [W. Cao, D. Li and Z. Zhang, Commun. Comput. Phys. 21 (1) (2017) 211-236]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L²-norm to (p+1)-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be p+2, when piecewise polynomials of degree at most p are used. We use these results to show that the leading error terms on each element for the solution and its derivative are proportional to (p+1)-degree right and left Radau polynomials. These new results enable us to construct residual-based a posteriori error estimates of the spatial errors. We further prove that, for smooth solutions, these a posteriori LDG error estimates converge, at a fixed time, to the true spatial errors in the L²-norm at O(h^{p+2}) rate. Finally, we show that the global effectivity indices in the L²-norm converge to unity at O(h) rate. The current results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be p+3 ⁄ 2 and 1 ⁄ 2, respectively. Our proofs are valid for arbitrary regular meshes using P^p polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results.

• Abstract

This paper is devoted to analyze of nonconforming finite volume methods (FVMs), whose trial spaces are chosen as the nonconforming finite element (FE) spaces, for solving the second order elliptic boundary value problems. We formulate the nonconforming FVMs as special types of Petrov-Galerkin methods and develop a general convergence theorem, which serves as a guide for the analysis of the nonconforming FVMs. As special examples, we shall present the triangulation based Crouzeix-Raviart (C-R) FVM as well as the rectangle mesh based hybrid Wilson FVM. Their optimal error estimates in the mesh dependent H¹-norm will be obtained under the condition that the primary mesh is regular. For the hybrid Wilson FVM, we prove that it enjoys the same optimal error order in the L²-norm as that of the Wilson FEM. Numerical experiments are also presented to confirm the theoretical results.

• Abstract

We present a novel numerical scheme to price European options on discount bond, where the single factor models are adopted for the short interest rate. This method is based on a fitted finite volume (FFVM) scheme for the spatial discretization and an implicit scheme for the time discretization. We show that this scheme is consistent, stable and monotone, hence it ensures the convergence to the solution of continuous problem. Numerical experiments are performed to verify the effectiveness and usefulness of this new method.

• Abstract

We present a general procedure to construct a non-linear mimetic finite-difference operator. The method is very simple and general: it can be applied for any order scheme, for any number of grid points and for any operator constraints. In order to validate the procedure, we apply it to a specific example, the Jacobian operator for the vorticity equation. In particular we consider a finite difference approximation of a second order Jacobian which uses a 9× 9 uniform stencil, verifies the skew-symmetric property and satisfies physical constraints such as conservation of energy and enstrophy. This particular choice has been made in order to compare the present scheme with Arakawa's renowned Jacobian, which turns out to be a specific case of the general solution. Other possible generalizations of Arakawa's Jacobian are available in literature but only the present approach ensures that the class of solutions found is the widest possible. A simplified analysis of the general scheme is proposed in terms of truncation error and study of the linearised operator together with some numerical experiments. We also propose a class of analytical solutions for the vorticity equation to compare an exact solution with our numerical results.

• Abstract

In this paper, we develop an optimal control approach of pollution and emission reduction subject to convection-diffusion transport equations. A linked simulation-optimization method has been proposed, based on solving the convection-diffusion transport equations and solving the optimization procedure. The governing equations of the convection-diffusion-reaction equations with pollution sources are discretized by the splitting improved upwind finite volume scheme while the constrained differential evolution (DE) algorithm is applied to solve the global optimization procedure. The advantage of the approach is the external linking of the numerical simulation and the optimization procedure, minimizing both the weighted deviation between simulated concentrations and the smallest allowable concentrations at observation sites and the emission reduction cost at the pollution sources at same time. Numerical tests first check the convergence of numerical methods. Numerical experiments then show the performance of the approach for solving the optimal control problems of pollution and cost of emission reduction. The developed optimal control approach is efficient and it can be applied to more complex problems in applications.

• Abstract

This article presents a mixed finite volume method for solving second-order elliptic equations with Neumann boundary conditions. The computational domains can be decomposed into non-overlapping sub-domains or blocks and the diffusion tensors may be discontinuous across the sub-domain boundaries. We define a conforming triangular partition on each sub-domains independently, and employ the standard mixed finite volume method within each sub-domain. On the interfaces between different sun-domains, the grids are non-matching. The Robin type boundary conditions are imposed on the non-matching interfaces to enhance the continuity of the pressure and flux. Both the solvability and the first order rate of convergence for this numerical scheme are rigorously proved. Numerical experiments are provided to illustrate the error behavior of this scheme and confirm our theoretical results.