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In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the number of terms used in the Fourier-cosine series expansion, where the error bound estimation is based on the characteristic function of the log-asset price process. A Matlab code implementing this technique is attached at the end of the paper. [ABSTRACT FROM AUTHOR]

This paper derives a posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem. We discuss a reconstruction in the standard H 0 1 {H_{0}^{1}} -conforming space for the primal variable of the mixed Laplace eigenvalue problem and compare it with analogous approaches present in the literature for the corresponding source problem. In the case of Raviart–Thomas finite elements of arbitrary polynomial degree, the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. Our reconstruction is performed locally on a set of vertex patches. [ABSTRACT FROM AUTHOR]

This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the L 2 {L^{2}} norm are derived. Numerical results are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of n-dimensional fluids, n ∈ { 2 , 3 } {n\in\{2,3\}} , with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported. [ABSTRACT FROM AUTHOR]

The Fourier pseudo-spectral method has been studied for a one- dimensional coupled system of viscous Burgers equations. Two test problems with known exact solutions have been selected for this study. In this paper, the rate of con- vergence in time and error analysis of the solution of the first problem has been studied, while the numerical results of the second problem obtained by the present method are compared to those obtained by using the Chebyshev spectral collocation method. The numerical results show that the proposed method outperforms the conventional one in terms of accuracy and convergence rate. [ABSTRACT FROM AUTHOR]

In this paper we give a new proof of the -error estimate for the finite element approximation of the Laplace-Beltrami equation with an additional lower order term on a surface. While the proof available in the literature uses the method of perturbed bilinear forms from Schatz and Wahlbin, we adapt Scott's proof from an Euclidean setting to the surface case. Furthermore, in contrast to the literature we use an approximative Green's function on the surface instead of an exact Green's function which is obtained by lifting an Euclidean Green's function locally from the tangent plane to the surface. [ABSTRACT FROM AUTHOR]

We consider linear ill-posed problems in Hilbert spaces with a noisy right hand side and a given noise level. To solve non-self-adjoint problems by the (iterated) Tikhonov method, one effective rule for choosing the regularization parameter is the monotone error rule (Tautenhahn&Hämarik, Inverse Problems, 1999, 15, 1487-1505). In this paper we consider the solution of self-adjoint problems by the (iterated) Lavrentiev method and propose for parameter choice an analog of the monotone error rule. We prove under certain mild assumptions the quasi-optimality of the proposed rule guaranteeing convergence and order optimal error estimates. Numerical examples show for the proposed rule and its modifications much better performance than for the modified discrepancy principle. [ABSTRACT FROM AUTHOR]

A robust finite element method is introduced for solving elastic vibration problems in two dimensions. The temporal discretization is carried out using the P 1 {P_{1}} -continuous discontinuous Galerkin (CDG) method, while the spatial discretization is based on the Crouziex–Raviart (CR) element. It is shown after a technical derivation that the error of the displacement (resp. velocity) in the energy norm (resp. L 2 {L^{2}} norm) is bounded by O ⁢ (h + k) {O(h+k)} (resp. O ⁢ (h 2 + k) {O(h^{2}+k)}), where h and k denote the mesh sizes of the subdivisions in space and time, respectively. Under some regularity assumptions on the exact solution, the error bound is independent of the Lamé coefficients of the elastic material under discussion. A series of numerical results are offered to illustrate numerical performance of the proposed method and some other fully discrete methods for comparison. [ABSTRACT FROM AUTHOR]

We perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. In order to obtain the quasi-optimal convergence incorporating second-order Runge–Kutta schemes for time discretization, we need a strengthened 4 / 3 {4/3} -CFL condition ( Δ ⁢ t ∼ h 4 / 3 {\Delta t\sim h^{4/3}}). To overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge–Kutta scheme for time discretization. We demonstrate the error estimates in L 2 {L^{2}} -sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree l ≥ 1 {l\geq 1} under the standard CFL condition. [ABSTRACT FROM AUTHOR]

An error estimate for the minimal error method for nonlinear ill-posed problems under general a Hölder-type source condition is not known. We consider a modified minimal error method for nonlinear ill-posed problems. Using a Hölder-type source condition, we obtain an optimal order error estimate. We also consider the modified minimal error method with noisy data and provide an error estimate. [ABSTRACT FROM AUTHOR]

We present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions. [ABSTRACT FROM AUTHOR]

We prove quasi-optimality of an adaptive finite element algorithm for a model problem of optimal control including control constraints. The quasi-optimility expresses the fact that the decrease of error with respect to the number of mesh cells is optimal up to a constant. The considered algorithm is based on an adaptive marking strategy which compares a standard residual-type a posteriori error estimator with a data approximation term in each step of the algorithm in order to adapt the marking of cells. [ABSTRACT FROM AUTHOR]

We study an efficient finite element method for the NS-ω model, that uses van Cittert approximate deconvolution to improve accuracy and Scott-Vogelius elements to provide pointwise mass conservative solutions and remove the dependence of the (often large) Bernoulli pressure error on the velocity error. We provide a complete numerical analysis of the method, including well-posedness, unconditional stability, and optimal convergence. Additionally, an improved choice of filtering radius (versus the usual choice of the average mesh width) for the scheme is found, by identifying a connection with a scheme for the velocity-vorticity-helicity NSE formulation. Several numerical experiments are given that demonstrate the performance of the scheme, and the improvement offered by the new choice of the filtering radius. [ABSTRACT FROM AUTHOR]