1,062 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
Robust globally divergence-free weak Galerkin finite element methods for natural convection problems
This paper proposes and analyzes a class of weak Galerkin (WG) finite element
methods for stationary natural convection problems in two and three dimensions.
We use piecewise polynomials of degrees k, k-1, and k(k>=1) for the velocity,
pressure, and temperature approximations in the interior of elements,
respectively, and piecewise polynomials of degrees l, k, l(l = k-1,k) for the
numerical traces of velocity, pressure and temperature on the interfaces of
elements. The methods yield globally divergence-free velocity solutions.
Well-posedness of the discrete scheme is established, optimal a priori error
estimates are derived, and an unconditionally convergent iteration algorithm is
presented. Numerical experiments confirm the theoretical results and show the
robustness of the methods with respect to Rayleigh number.Comment: 32 pages, 13 figure
An Weak Galerkin FEM for singularly perturbed problems
We present the analysis for an weak Galerkin-FEM for singularly
perturbed reaction-convection-diffusion problems in one-dimension. Under the
analyticity of the data assumption, we establish robust exponential
convergence, when the error is measured in the energy norm, as the degree
of the approximating polynomials is increased. The Spectral Boundary Layer mesh
is used, which is the minimal (layer adapted) mesh for such problems. Numerical
examples illustrating the theory are also presented
Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements
The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical
Methods for Singularly Perturbed Differential Equations" appeared many years
ago and was for many years a reliable guide into the world of numerical methods
for singularly perturbed problems. Since then many new results came into the
game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827
A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients
In this paper, a weak Galerkin finite element method for solving the time fractional reaction-convection diffusion problem is proposed. We use the well known L1 discretization in time and a weak Galerkin finite element method on uniform mesh in space. Both continuous and discrete time weak Galerkin finite element method are considered and analyzed. The stability of the discrete time scheme is proved. The error estimates for both schemes are given. Finally, we give some numerical experiments to show the efficiency of the proposed method
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