Page 451 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 451
MODELING, APPROXIMATION, AND ANALYSIS OF PARTIAL DIFFERENTIAL
EQUATIONS INVOLVING SINGULAR SOURCE TERMS (MS-39)
a priori approximation theory, which requires to develop the stability of the Stokes projector
over weighted spaces. In the case that the forcing is a linear combination of Dirac deltas, we
develop a posteriori error estimators for the stationary Stokes and Navier Stokes problems. We
show that our estimators are reliable and locally efficient, and illustrate their performance within
an adaptive method. We briefly comment on ongoing work regarding the Bousinessq system.
Numerical experiments illustrate and complement our theory.
Discontinuous Galerkin Discretisations for Problems with Dirac Delta
Source
Thomas Wihler, wihler@math.unibe.ch
University of Bern, Switzerland
Coauthor: Paul Houston
We investigate the symmetric interior penalty discontinuous Galerkin (SIPG) scheme for the
numerical approximation of linear second-order elliptic PDE with Dirac delta right-hand side.
We outline both an a priori and a (residual-type) a posteriori error analysis on the error measured
in terms of the L2–norm. Moreover, some computational results will be presented. Finally, a
brief outlook on an inf-sup theory in weighted Sobolev spaces will be given.
Regularity and finite element approximation for two-dimensional elliptic
equations with line Dirac sources
Peimeng Yin, pyin@wayne.edu
Wayne State University, United States
We study the elliptic equation with a line Dirac delta function as the source term subject to the
Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes
different types of solution singularities in the neighborhood of the line fracture. We establish
new regularity results for the solution in a class of weighted Sobolev spaces and propose fi-
nite element algorithms that approximate the singular solution at the optimal convergence rate.
Numerical tests are presented to justify the theoretical findings.
Analysis and approximation of mixed-dimensional PDEs on 3D-1D
domains coupled with Lagrange multipliers
Paolo Zunino, paolo.zunino@polimi.it
Politecnico di Milano, Italy
Coauthors: Miroslav Kuchta, Kent-Andre Mardal
Coupled partial differential equations (PDEs) defined on domains with different dimensionality
are usually called mixed-dimensional PDEs. We address mixed-dimensional PDEs on three-
dimensional (3D) and one-dimensional (1D) domains, which gives rise to a 3D-1D coupled
problem. Such a problem poses several challenges from the standpoint of existence of solutions
and numerical approximation. For the coupling conditions across dimensions, we consider the
combination of essential and natural conditions, which are basically the combination of Dirich-
449
EQUATIONS INVOLVING SINGULAR SOURCE TERMS (MS-39)
a priori approximation theory, which requires to develop the stability of the Stokes projector
over weighted spaces. In the case that the forcing is a linear combination of Dirac deltas, we
develop a posteriori error estimators for the stationary Stokes and Navier Stokes problems. We
show that our estimators are reliable and locally efficient, and illustrate their performance within
an adaptive method. We briefly comment on ongoing work regarding the Bousinessq system.
Numerical experiments illustrate and complement our theory.
Discontinuous Galerkin Discretisations for Problems with Dirac Delta
Source
Thomas Wihler, wihler@math.unibe.ch
University of Bern, Switzerland
Coauthor: Paul Houston
We investigate the symmetric interior penalty discontinuous Galerkin (SIPG) scheme for the
numerical approximation of linear second-order elliptic PDE with Dirac delta right-hand side.
We outline both an a priori and a (residual-type) a posteriori error analysis on the error measured
in terms of the L2–norm. Moreover, some computational results will be presented. Finally, a
brief outlook on an inf-sup theory in weighted Sobolev spaces will be given.
Regularity and finite element approximation for two-dimensional elliptic
equations with line Dirac sources
Peimeng Yin, pyin@wayne.edu
Wayne State University, United States
We study the elliptic equation with a line Dirac delta function as the source term subject to the
Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes
different types of solution singularities in the neighborhood of the line fracture. We establish
new regularity results for the solution in a class of weighted Sobolev spaces and propose fi-
nite element algorithms that approximate the singular solution at the optimal convergence rate.
Numerical tests are presented to justify the theoretical findings.
Analysis and approximation of mixed-dimensional PDEs on 3D-1D
domains coupled with Lagrange multipliers
Paolo Zunino, paolo.zunino@polimi.it
Politecnico di Milano, Italy
Coauthors: Miroslav Kuchta, Kent-Andre Mardal
Coupled partial differential equations (PDEs) defined on domains with different dimensionality
are usually called mixed-dimensional PDEs. We address mixed-dimensional PDEs on three-
dimensional (3D) and one-dimensional (1D) domains, which gives rise to a 3D-1D coupled
problem. Such a problem poses several challenges from the standpoint of existence of solutions
and numerical approximation. For the coupling conditions across dimensions, we consider the
combination of essential and natural conditions, which are basically the combination of Dirich-
449
