A higher‐order accurate Petrov–Galerkin finite‐element method for elliptic boundary‐value problems

Abstract

SUMMARYWe formulate a higher-order (superconvergent) Petrov-Galerkin method by determining, using a finitedifference approximation, the optimal selection of quadratic and cubic modifications to the standard linear test function for bilinear elements. Application of this method to linear elliptic problems results in improved accuracy and higher rates of convergence for problems with constant coefficients and improved accuracy for problems with variable coefficients. Supporting numerical examples are given.

Compact high‐order finite‐element method for elliptic transport problems with variable coefficients

“…(7)- (9)J. This approach has previously been shown to yield superconvergent solutions to the special case of (1) where c and d are constants [15]. To achieve superconvergent approximations to problems involving variable c and d it is also necessary, as in the one-dimensional case, to do the following: (a) explicitly add artificial diffusion and (b) treat coefficients ukl, a m , and Pm as local constants evaluated at each grid point A straightforward extension of the previous one-dimensional analysis yields the follow-…”

“…( l l ) ] can be written from Eqs. (3)-(9) as We next examine the application of approximation (15) to problems involving constant and variable convective coefficient c.…”

“…In a previous study [15] we determined that for the following choice of coefficients, PG approximation (15) yields an 0 ( h 4 ) difference approximation to (13) at each interior grid point x,. Accordingly, introducing (16) into (15) and integrating, we obtain the 0 ( h 4 ) CHOFE approximation…”

“…(7)-(9)J. This approach has previously been shown to yield superconvergent solutions to the special case of (1) where c and d are constants [15]. To achieve superconvergent approximations to problems involving variable c and d it is also necessary, as in the one-dimensional case, to do the following: In this study we restrict our attention to the Dirichlet problem.…”

“…Exact and linear Galerkin finite-element solutions to the homogeneous 1D convec-B. Variable Convective CoefficientFor problems involving variable c ( x ) , we employ a slightly nonstandard treatment of the integrand in(15) by treating coefficients a l : ( x ) , a , ( x ) , and p x ( x )as local constants evaluated at each x i . That is, tion-diffusion problem.Exact and compact high-order finite-element solutions to the homogeneous 1D convec-…”