3 research outputs found
First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems
We present and analyze a first order least squares method for convection
dominated diffusion problems, which provides robust L2 a priori error estimate
for the scalar variable even if the given data f in L2 space. The novel
theoretical approach is to rewrite the method in the framework of discontinuous
Petrov - Galerkin (DPG) method, and then show numerical stability by using a
key equation discovered by J. Gopalakrishnan and W. Qiu [Math. Comp. 83(2014),
pp. 537-552]. This new approach gives an alternative way to do numerical
analysis for least squares methods for a large class of differential equations.
We also show that the condition number of the global matrix is independent of
the diffusion coefficient. A key feature of the method is that there is no
stabilization parameter chosen empirically. In addition, Dirichlet boundary
condition is weakly imposed. Numerical experiments verify our theoretical
results and, in particular, show our way of weakly imposing Dirichlet boundary
condition is essential to the design of least squares methods - numerical
solutions on subdomains away from interior layers or boundary layers have
remarkable accuracy even on coarse meshes, which are unstructured
quasi-uniform
Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation
This paper presents an a-posteriori goal-oriented error analysis for a
numerical approximation of the steady Boltzmann equation based on a
moment-system approximation in velocity dependence and a discontinuous Galerkin
finite-element (DGFE) approximation in position dependence. We derive
computable error estimates and bounds for general target functionals of
solutions of the steady Boltzmann equation based on the DGFE moment
approximation. The a-posteriori error estimates and bounds are used to guide a
model adaptive algorithm for optimal approximations of the goal functional in
question. We present results for one-dimensional heat transfer and shock
structure problems where the moment model order is refined locally in space for
optimal approximation of the heat flux.Comment: arXiv admin note: text overlap with arXiv:1602.0131