Optimal Order Convergence Rates Research Articles

Numerical solution for a sub-diffusion equation with a smooth kernel

In this paper we study the numerical solution of an initial value problem of a sub-diffusion type. For the time discretization we apply the discontinuous Galerkin method and we use continuous piecewise finite elements for the space discretization. Optimal order convergence rates of our numerical solution have been shown. We compare our theoretical error bounds with the results of numerical computations. We also present some numerical results showing the super-convergence rates of the proposed method.

An Optimal Order Error Analysis of the One-Dimensional Quasicontinuum Approximation

We derive a model problem for quasicontinuum approximations that allows a simple, yet insightful, analysis of the optimal-order convergence rate in the continuum limit for both the energy-based quasicontinuum approximation and the quasi-nonlocal quasicontinuum approximation. For simplicity, the analysis is restricted to the case of second-neighbor interactions and is linearized about a uniformly stretched reference lattice. The optimal-order error estimates for the quasi-nonlocal quasicontinuum approximation are given for all strains up to the continuum limit strain for fracture. The analysis is based on an explicit treatment of the coupling error at the atomistic-to-continuum interface, combined with an analysis of the error due to the atomistic and continuum schemes using the stability of the quasicontinuum approximation.

A NEW NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY

We have developed new nonconforming mixed finite element methods for linear elasticity with a pure traction (displacement) boundary condition based on the Hellinger–Reissner variational principle using rectangular elements. Convergence analysis yields an optimal (suboptimal) convergence rate of [Formula: see text] for the L2-error of the stress and [Formula: see text] for the displacement in the pure traction (displacement) boundary problem. However, numerical experiments have yielded optimal-order convergence rates for both stress and displacement in both problems and have shown superconvergence for the displacement at the midpoint of each element. Moreover, we observed that the optimal convergence rates are still valid for large λ.

Numerical Integration over Spheres of Arbitrary Dimension

In this paper, we study the worst-case error (of numerical integration) on the unit sphere $\\mathbb{S}^{d}$, $d\\geq 2$, for all functions in the unit ball of the Sobolev space $\\mathbb{H}^s(\\mathbb{S}^d)$, where $s>d/2$. More precisely, we consider infinite sequences $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of $m(n)$-point numerical integration rules $Q_{m(n)}$, where (i) $Q_{m(n)}$ is exact for all spherical polynomials of degree $\\leq n$, and (ii) $Q_{m(n)}$ has positive weights or, alternatively to (ii), the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$ satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration) $E(Q_{m(n)};\\mathbb{H}^s(\\matbb{S}^d))$ in $\\mathbb{H}^s(\\mathbb{S}^d)$ has the upper bound $c n^{-s}$, where the constant $c$ depends on $s$ and $d$ (and possibly the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$). This extends the recent results for the sphere $\\mathbb{S}^2$ by K.Hesse and I.H.Sloan to spheres $\\mathbb{S}^d$ of arbitrary dimension $d\\geq2$ by using an alternative representation of the worst-case error. If the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of numerical integration rules satisfies $m(n)=\\mathcal{O}(n^d)$ an order-optimal rate of convergence is achieved.

A Lepskij-type stopping rule for regularized Newton methods

We investigate an a posteriori stopping rule of Lepskij-type for a class of regularized Newton methods and show that it leads to order optimal convergence rates for Hölder and logarithmic source conditions without a priori knowledge of the smoothness of the solution. Numerical experiments show that this stopping rule yields results at least as good as, and in some situations significantly better than, Morozov's discrepancy principle.

Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations

We consider space-time continuous Galerkin methods with mesh modification in time for semilinear second order hyperbolic equations. We show a priori estimates in the energy norm without mesh conditions. Under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates. Estimates of the jump in the Riesz projection in two successive time steps are also derived.

Tikhonov regularization and a posteriori rules for solving nonlinear ill posedproblems

Besides a priori parameter choice we study a posteriori rules for choosing the regularizationparameter αin the Tikhonov regularization method for solving nonlinear ill posed problemsF (x) = y,namely a rule 1 of Scherzer et al (Scherzer O, Engl H W and Kunisch K 1993 SIAM J.Numer. Anal. 30 1796–838) and a new rule 2 which is a generalization of themonotone error rule of Tautenhahn and Hämarik (Tautenhahn U and Hämarik U 1999Inverse Problems 15 1487–505) to the nonlinear case. We suppose that instead ofy there are givennoisy data yδsatisfying |y − yδ| ≤ δ with known noiselevel δand prove that rule 1 and rule 2 yield order optimal convergence rates O(δp/(p+1)) for theranges p ∊ (0, 2]and p ∊ (0, 1],respectively. Compared with foregoing papers our order optimal convergence rateresults have been obtained under much weaker assumptions which is important inengineering practice. Numerical experiments verify some of the theoretical results.

Symmetric mixed covolume methods for parabolic problems

AbstractWe present a mixed covolume method for a system of first order partial differential equations resulting from the mixed formulation of the general self‐adjoint parabolic problem with a variable nondiagonal diffusion tensor. The lowest order Raviart‐Thomas mixed element space on rectangles is used. We prove the first order optimal rate of convergence for approximate pressure as well as for approximate velocity. We also prove the second order superconvergence both for approximate velocity and pressure in certain discrete norms. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 561–583, 2002

Mixed Covolume Methods on Rectangular Grids For Elliptic Problems

We consider a covolume method for a system of first order PDEs resulting from the mixed formulation of the variable-coefficient-matrix Poisson equation with the Neumann boundary condition. The system may be used to represent the Darcy law and the mass conservation law in anisotropic porous media flow. The velocity and pressure are approximated by the lowest order Raviart--Thomas space on rectangles. The method was introduced by Russell [Rigorous Block-centered Discretizations on Irregular Grids: Improved Simulation of Complex Reservoir Systems, Reservoir Simulation Research Corporation, Denver, CO, 1995] as a control-volume mixed method and has been extensively tested by Jones [A Mixed Finite Volume Elementary Method for Accurate Computation of Fluid Velocities in Porous Media, University of Colorado at Denver, 1995] and Cai et al. [Computational Geosciences, 1 (1997), pp. 289--345]. We reformulate it as a covolume method and prove its first order optimal rate of convergence for the approximate velocities as well as for the approximate pressures.

Runge-Kutta characteristic methods for first-order linear hyperbolic equations

We develop two Runge–Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian–Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov–Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge–Kutta methods developed in this article. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 617–661, 1997

Runge–Kutta characteristic methods for first‐order linear hyperbolic equations

We develop two Runge–Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian–Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov–Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge–Kutta methods developed in this article. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 617–661, 1997

Optimal-order convergence rates for Eulerian-Lagrangian localized adjoint methods for reactive transport and contamination in groundwater : Hong Wang & R. E. Ewing, Numerical Methods for Partial Differential Equations, 11(1), 1995, pp 1–31

Optimal-order convergence rates for Eulerian-Lagrangian localized adjoint methods for reactive transport and contamination in groundwater : Hong Wang & R. E. Ewing, Numerical Methods for Partial Differential Equations, 11(1), 1995, pp 1–31

An Optimal-Order Estimate for Eulerian–Lagrangian Localized Adjoint Methods for Variable-Coefficient Advection-Reaction Problems

Previously, we developed Eulerian–Lagrangian localized adjoint methods (ELLAMs) to solve advection-reaction problems. ELLAMs provide a systematic approach to treat boundary conditions and to conserve mass and have proven to be powerful methods for advection-dominated problems. In this paper, the authors conduct theoretical analysis for these ELLAMs and prove that they have an optimal-order convergence rate. Moreover, the estimates involve only the derivatives of the exact solution in space and along the characteristics. In contrast, many existing methods have only suboptimal-order error estimates for advection-reaction problems and the estimates involve the temporal derivatives of the exact solution, which are usually much larger than the derivatives along the characteristics.Key words. advection-reaction equations, Eulerian–Lagrangian method, convergence analysis, optimal-order error estimates

The minimal error conjugate gradient method is a regularization method

The regularizing properties of the conjugate gradient iteration, applied to the normal equation of a linear ill-posed problem, were established by Nemirovskii in 1986. A seemingly more attractive variant of this algorithm is the minimal error method suggested by King. The present paper analyzes the regularizing properties of the minimal error method. It is shown that the discrepancy principle is no regularizing stopping rule for the minimal error method. Instead, a different stopping rule is suggested which leads to order-optimal convergence rates.

Alternating Direction Multistep Methods for Parabolic Problems–Iterative Stabilization

Efficient multistep procedures for time-stepping Galerkin methods for parabolic partial differential equations are presented and analyzed. The procedures involve using an alternating direction preconditioned iterative method for approximately solving the linear equations arising at each timestep in a discrete Galerkin method. Optimal order convergence rates are obtained for the iterative methods. Work estimates of almost optimal order are obtained.

Efficient multistep procedures for nonlinear parabolic problems with nonlinear Neumann boundary conditions

Efficient multistep procedures for time-stepping Galerkin methods for nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions are presented and analyzed. The procedures involve using a precoditioned iterative method for approximately solving the differet linear equations arising at each time step in a discrete time Galerkin method. Optimal order convergence rates are obtained for the iterative methods. Work estimates of almost optimal order are obtained.

Efficient Time-Stepping Methods for Miscible Displacement Problems in Porous Media

Efficient procedures for time-stepping Galerkin methods for approximating the solution of a coupled system for $c = c(x,t)$ and $p = p(x,t)$, with nonlinear Neumann boundary conditions, of the form \[ \begin{gathered} - \nabla \cdot [a(x,c)\{ \nabla p - y(x,c)\nabla g\} ] \equiv \nabla \cdot u = f_1 (x,t),\quad x \in \Omega ,\quad t \in (0,T], \hfill \\ \nabla \cdot [b(x,c,\nabla p)\nabla c] - u \cdot \nabla c = \phi (x)\frac{{\partial c}} {{\partial t}} - f_2 (x,t,c),\quad x \in \Omega ,\quad t \in (0,T], \hfill \\ u \cdot v = q_1 (x,t),\quad x \in \partial \Omega ,\quad t \in (0,T], \hfill \\ b\frac{{\partial c}} {{\partial v}} = q(x,t,c),\quad x \in \partial \Omega ,\quad t \in (0,T], \hfill \\ c(x,0) = c_0 (x),\quad x \in \Omega , \hfill \\ \end{gathered} \] where $\Omega \subset \mathbb{R}^d $, $2 \leqq d \leqq 3$, are presented and analyzed. This system is a possible model system for describing the miscible displacement of one incompressible fluid by another in a porous medium when flow conditions are prescribed on the boundary. The procedures involve the use of a preconditioned iterative method for approximately solving the algebraic problem at each time step. The iteration need be performed only long enough to stabilize the scheme. Motivated by the fact that the pressure is smoother in time than the concentration, larger time steps are used for the pressure than for the concentration. Under certain smoothness assumptions on the solution, optimal order convergence rates and almost optimal order work estimates are obtained.