TY - JOUR
T1 - The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces
AU - Muga, Ignacio
AU - Tyler, Matthew J.W.
AU - Van Der Zee, Kristoffer G.
N1 - Funding Information:
Funding: The work by Ignacio Muga was done in the framework of Chilean FONDECYT research project No. 1160774. Ignacio Muga was also partially supported by the European Union’s Horizon 2020, research and innovation program under the Marie Sklodowska-Curie grant agreement No. 777778. Matthew Tyler and Kristoffer van der Zee are grateful for the support provided by the London Mathematical Society (LMS) Undergraduate Research Bursary Grant “Advanced discontinuous discretisation techniques for multiscale partial differential equations” 17-18 103, and thank Donald Brown for his contributions. Kristoffer van der Zee also thanks the support provided by the Royal Society International Exchanges Scheme/Kan Tong Po Visiting Fellowship Programme, and the above FONDECYT project.
Publisher Copyright:
© 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.
PY - 2019/7/1
Y1 - 2019/7/1
N2 - We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue Lp, 1 < p < ∞. The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in L p {L^{p}}, and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.
AB - We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue Lp, 1 < p < ∞. The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in L p {L^{p}}, and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.
KW - Advection-Reaction
KW - Banach Spaces
KW - Compatible FE Pairs
KW - DDMRes
KW - Discrete Dual Norms
KW - Fortin Condition
KW - Petrov-Galerkin Method
KW - Residual Minimization
UR - http://www.scopus.com/inward/record.url?scp=85068408938&partnerID=8YFLogxK
U2 - 10.1515/cmam-2018-0199
DO - 10.1515/cmam-2018-0199
M3 - Article
AN - SCOPUS:85068408938
VL - 19
SP - 557
EP - 579
JO - Computational Methods in Applied Mathematics
JF - Computational Methods in Applied Mathematics
SN - 1609-4840
IS - 3
ER -