Adjoint Recovery of Superconvergent Linear Functionals from Galerkin Approximations. The One-dimensional Case

نویسندگان

  • Bernardo Cockburn
  • Ryuhei Ichikawa
چکیده

In this paper, we extend the adjoint error correction of Pierce and Giles [SIAM Review, 42 (2000), pp. 247-264] for obtaining superconvergent approximations of functionals to Galerkin methods. We illustrate the technique in the framework of discontinuous Galerkin methods for ordinary differential and convection-diffusion equations in one space dimension. It is well known that approximations to linear functionals obtained by discontinuous Galerkin methods with polynomials of degree k can be proven to converge with order 2 k + 1 and 2 k for ordinary differential and convection-diffusion equations, respectively. In contrast, the order of convergence of the adjoint error correction method can be proven to be 4 k+1 and 4 k, respectively. Since both approaches have a computational complexity of the same order, the adjoint error correction method is clearly a competitive alternative. Numerical results which confirm the theoretical predictions are presented.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Adjoint Recovery of Superconvergent Functionals from PDE Approximations

Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding glob...

متن کامل

Adjoint Recovery of Superconvergent Functionals from Approximate Solutions of Partial Diierential Equations

Motivated by applications in computational uid dynamics, we present a method for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed ow solution on which they are based. This is achieved through error analysis which uses an ad-joint p.d.e. to relate the local errors in approximating the ow solution to the corresponding global ...

متن کامل

Some a Priori Error Estimates for Finite Element Approximations of Elliptic and Parabolic Linear Stochastic Partial Differential Equations

We study some theoretical aspects of Legendre polynomial chaos based finite element approximations of elliptic and parabolic linear stochastic partial differential equations (SPDEs) and provide a priori error estimates in tensor product Sobolev spaces that hold under appropriate regularity assumptions. Our analysis takes place in the setting of finitedimensional noise, where the SPDE coefficien...

متن کامل

Error estimation and adjoint-based refinement for multiple force coefficients in aerodynamic flow simulations

In this talk we give an overview of recent developments on adaptive higher order Discontinuous Galerkin discretizations for the use in computational aerodynamics at the DLR in Braunschweig. In particular, this includes some of the most recent developments and results achieved in the EU project ADIGMA. Important quantities of interest in aerodynamic flow simulations are the aerodynamic force coe...

متن کامل

Analysis of Optimal Superconvergence of Local Discontinuous Galerkin Method for One-dimensional Linear Parabolic Equations

In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when alternating flux is used. We prove that if we apply piecewise k-th degree polynomials, the error between the LDG solution and the exact solution is (k + 2)-th order superconvergent at the Radau points with suitable initial...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • J. Sci. Comput.

دوره 32  شماره 

صفحات  -

تاریخ انتشار 2007