Applied Numerical Mathematics
Even-odd goal-oriented a posteriori error estimation for elliptic problems
Applied Numerical Mathematics
Journal of Computational and Applied Mathematics
Extrapolation discontinuous Galerkin method for ultraparabolic equations
Journal of Computational and Applied Mathematics
On the valuation of interest rate products under multi-factor HJM term-structures
Applied Numerical Mathematics
Even--odd goal-oriented a posteriori error estimation for elliptic problems
Applied Numerical Mathematics
An adaptive extrapolation discontinuous Galerkin method for the valuation of Asian options
Journal of Computational and Applied Mathematics
On Adaptive Eulerian---Lagrangian Method for Linear Convection---Diffusion Problems
Journal of Scientific Computing
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Adjerid, Babuska, and Flaherty [ Math. Models Methods Appl. Sci., 9 (1999), pp. 261--286] and Yu [Math. Numer. Sinica, 13 (1991), pp. 89--101] and [Math. Numer. Sinica, 13 (1991), pp. 307--314] show that a posteriori estimates of spatial discretization errors of piecewise bi- p polynomial finite element solutions of elliptic and parabolic problems on meshes of square elements may be obtained from jumps in solution gradients at element vertices when p is odd and from local elliptic or parabolic problems when p is even. We show that these simple error estimates are asymptotically correct for other finite element spaces. The key requirement is that the trial space contain all monomial terms of degree p + 1 except for $ x_1^{p+1}$ and $ x_2^{p+1}$ in a Cartesian (x1,x2 ) frame. Computational results show that the error estimates are accurate, robust, and efficient for a wide range of problems, including some that are not supported by the present theory. These involve quadrilateral-element meshes, problems with singularities, and nonlinear problems.