55 research outputs found
An analysis of the practical DPG method
In this work we give a complete error analysis of the Discontinuous Petrov
Galerkin (DPG) method, accounting for all the approximations made in its
practical implementation. Specifically, we consider the DPG method that uses a
trial space consisting of polynomials of degree on each mesh element.
Earlier works showed that there is a "trial-to-test" operator , which when
applied to the trial space, defines a test space that guarantees stability. In
DPG formulations, this operator is local: it can be applied
element-by-element. However, an infinite dimensional problem on each mesh
element needed to be solved to apply . In practical computations, is
approximated using polynomials of some degree on each mesh element. We
show that this approximation maintains optimal convergence rates, provided that
, where is the space dimension (two or more), for the Laplace
equation. We also prove a similar result for the DPG method for linear
elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods
are also included.Comment: Mathematics of Computation, 201
An Analysis of the Practical DPG Method
We give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree p on each mesh element. Earlier works showed that there is a trial-to-test operator T, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator T is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply T. In practical computations, T is approximated using polynomials of some degree r \u3e p on each mesh element. We show that this approximation maintains optimal convergence rates, provided that r p + N, where N is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included
Convergence rates of the DPG method with reduced test space degree
This paper presents a duality theorem of the Aubin-Nitsche type for
discontinuous Petrov Galerkin (DPG) methods. This explains the numerically
observed higher convergence rates in weaker norms. Considering the specific
example of the mild-weak (or primal) DPG method for the Laplace equation, two
further results are obtained. First, the DPG method continues to be solvable
even when the test space degree is reduced, provided it is odd. Second, a
non-conforming method of analysis is developed to explain the numerically
observed convergence rates for a test space of reduced degree
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
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