On the Convergence Rate Estimates for Finite Difference Schemes Approximating Homogeneous Initial-boundary Value Problem for Hyperbolic Equation
On the derivation of the modified equation for the analysis of linear numerical methods
On the strong stability of operator-difference schemes in time-integral norms.
On the Structure of Error Estimates for Finite-Difference Methods.
On the Transport-Diffusion Algorithm and Its Applications to the NavierStokes Equations.
Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations
We consider the original DG method for solving the advection-reaction equations with arbitrary velocity in space dimensions. For triangulations satisfying the flow condition, we first prove that the optimal convergence rate is of order in the -norm if the method uses polynomials of order . Then, a very simple derivative recovery formula is given to produce an approximation to the derivative in the flow direction which superconverges with order . Further we consider a residual-based a posteriori...
Optimal error estimates for semidiscrete phase relaxation models
Option valuation under the VG process by a DG method
The paper presents a discontinuous Galerkin method for solving partial integro-differential equations arising from the European as well as American option pricing when the underlying asset follows an exponential variance gamma process. For practical purposes of numerical solving we introduce the modified option pricing problem resulting from a localization to a bounded domain and an approximation of small jumps, and we discuss the related error estimates. Then we employ a robust numerical procedure...
Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem
Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem
We consider the analysis and numerical solution of a forward-backward boundary value problem. We provide some motivation, prove existence and uniqueness in a function class especially geared to the problem at hand, provide various energy estimates, prove a priori error estimates for the Galerkin method, and show the results of some numerical computations.
Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation
We present in this paper a proof of well-posedness and convergence for the parallel Schwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heat equation. Since the equation we study is an evolution one, each subproblem at each step has its own local existence time, we then determine a common existence time for every problem in any subdomain at any step. We also introduce a new technique: Exponential Decay Error Estimates, to prove the convergence of the Schwarz Methods, with...
Rate of convergence of finite-difference approximations for degenerate linear parabolic equations with and coefficients.
Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: Error estimates
In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain , with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant , accompanied with a nonlocal (integral) Dirichlet side condition. We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution and...
Reduced basis method for finite volume approximations of parametrized linear evolution equations
The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations....
Resolvent estimates of elliptic differential and finite-element operators in pairs of function spaces.
Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable -scheme with . Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo 40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization...
Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization...
Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes
Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction–diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element...
Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes
Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element...
Rothe time-discretization method for a nonlocal problem arising in thermoelasticity.