573 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
Partial expansion of a Lipschitz domain and some applications
We show that a Lipschitz domain can be expanded solely near a part of its
boundary, assuming that the part is enclosed by a piecewise C1 curve. The
expanded domain as well as the extended part are both Lipschitz. We apply this
result to prove a regular decomposition of standard vector Sobolev spaces with
vanishing traces only on part of the boundary. Another application in the
construction of low-regularity projectors into finite element spaces with
partial boundary conditions is also indicated
A first order system least squares method for the Helmholtz equation
We present a first order system least squares (FOSLS) method for the
Helmholtz equation at high wave number k, which always deduces Hermitian
positive definite algebraic system. By utilizing a non-trivial solution
decomposition to the dual FOSLS problem which is quite different from that of
standard finite element method, we give error analysis to the hp-version of the
FOSLS method where the dependence on the mesh size h, the approximation order
p, and the wave number k is given explicitly. In particular, under some
assumption of the boundary of the domain, the L2 norm error estimate of the
scalar solution from the FOSLS method is shown to be quasi optimal under the
condition that kh/p is sufficiently small and the polynomial degree p is at
least O(\log k). Numerical experiments are given to verify the theoretical
results
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