TY - JOUR
T1 - An analysis of the practical DPG method
AU - Gopalakrishnan, J.
AU - Qiu, W.
PY - 2014/3
Y1 - 2014/3
N2 - Abstract. 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 ≥ 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. © 2013 American Mathematical Society.
AB - Abstract. 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 ≥ 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. © 2013 American Mathematical Society.
KW - Discontinuous galerkin
KW - DPG method
KW - Petrov-Galerkin
KW - Ultraweak formulation
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U2 - 10.1090/S0025-5718-2013-02721-4
DO - 10.1090/S0025-5718-2013-02721-4
M3 - 21_Publication in refereed journal
VL - 83
SP - 537
EP - 552
JO - Mathematics of Computation
JF - Mathematics of Computation
SN - 0025-5718
IS - 286
ER -