2 research outputs found
A Framework for the Numerical Computation and a Posteriori Verification of Invariant Objects of Evolution Equations
We develop a theoretical framework for computer-assisted proofs of the
existence of invariant objects in semilinear PDEs. The invariant objects
considered in this paper are equilibrium points, traveling waves, periodic
orbits and invariant manifolds attached to fixed points or periodic orbits. The
core of the study is writing down the invariance condition as a zero of an
operator. These operators are in general not continuous, so one needs to smooth
them by means of preconditioners before classical fixed point theorems can be
applied. We develop in detail all the aspects of how to work with these
objects: how to precondition the equations, how to work with the nonlinear
terms, which function spaces can be useful, and how to work with them in a
computationally rigorous way. In two companion papers, we present two different
implementations of the tools developed in this paper to study periodic orbits.Comment: 17 page
Adaptive staggered DG method for Darcy flows in fractured porous media
Modeling flows in fractured porous media is important in applications. One
main challenge in numerical simulation is that the flow is strongly influenced
by the fractures, so that the solutions typically contain complex features,
which require high computational grid resolutions. Instead of using uniformly
fine mesh, a more computationally efficient adaptively refined mesh is
desirable. In this paper we design and analyze a novel residual-type a
posteriori error estimator for staggered DG methods on general polygonal meshes
for Darcy flows in fractured porous media. The method can handle fairly general
meshes and hanging nodes can be simply incorporated into the construction of
the method, which is highly appreciated for adaptive mesh refinement. The
reliability and efficiency of the error estmator are proved. The derivation of
the reliability hinges on the stability of the continuous setting in the primal
formulation. A conforming counterpart that is continuous within each bulk
domain for the discrete bulk pressure is defined to facilitate the derivation
of the reliability. Finally, several numerical experiments including multiple
non-intersecting fractures are carried out to confirm the proposed theories.Comment: 20 pages, 16 figures. arXiv admin note: text overlap with
arXiv:2005.1095