Métodos miméticos de pasos fraccionarios - E-Archivo - Universidad ...

6 introduction Ch. 1
resulting algorithm (cf. Chapter 5).
The rest of the thesis is organized as follows. In Chapter 2, we consider a semilinear
problem of type (1.1) with homogeneous Dirichlet boundary conditions.
This chapter is mainly concerned with computational issues and is intended to
be an introduction to mimetic fractional step methods for subsequent chapters.
In order to obtain a numerical approximation to the solution of the original
problem, we combine a cell-node MFD method with an FSRKm+1 scheme. In
particular, the mimetic technique considers the degrees of freedom for ψ to be located
at the geometric centers of the cells, while assuming those for the Cartesian
components of u on the mesh nodes. As for the time integration, we introduce a
domain decomposition operator splitting in combination with a linearly implicit
variant of the well-known Yanenko’s method (cf. Yanenko (1971)). The fully
discrete scheme provides unconditionally stable solutions which are tested to be
second-order convergent in space and first-order convergent in time. The numerical
results further illustrate the behaviour of the algorithm on a collection
of logically rectangular grids.
Based on the preceding discretization techniques, Chapter 3 establishes a
suitable theoretical framework which permits to develop the convergence analysis
for both the semidiscrete and totally discrete schemes. In this case, we
consider the semilinear problem of the previous chapter with homogeneous Neumann
boundary conditions. Again, a mimetic fractional step method aims to be
constructed for the solution of such a problem. Regarding the spatial semidiscretization,
we introduce a cell-edge MFD scheme whose degrees of freedom for
ψ are again defined at the cell centers, while those for u are chosen to be located
at the midpoints of the mesh edges –in fact, they provide an approximation to
the orthogonal projections of u onto the unit vectors normal to the mesh edges.
Note that such degrees of freedom are precisely those of the lowest order Raviart–
Thomas quadrilateral element. This property permits to derive an equivalence
between the MFD scheme and a suitable MFE method, which will be the key to
formulating the convergence analysis of the former within the abstract setting
of the latter. On the other hand, we adapt the domain decomposition splitting
of the preceding chapter to the newly defined discrete diffusion operator, and
further combine it with a family of linearly implicit FSRKm+1 methods inspired
in the Peaceman–Rachford ADI scheme. The resulting algorithm is shown to
be second-order convergent in both space and time, while preserving its unconditional
stability. To conclude, the theoretical results are illustrated by some
numerical examples.
Thus far, the problems under consideration are posed on polygonal domains
which are likely to be covered with a logically rectangular mesh. In Chapter 4, we
consider physical domains of general geometry which admit a smooth transformation
into a rectangular computational domain. In this case, (1.1) is endowed
with non-homogeneous Dirichlet boundary conditions. After the original problem
is mapped onto the computational domain, we apply a proper combination
of a CCFD scheme with an FSRKm+1 method based on a component-wise splitting.
More precisely, we first introduce an expanded MFE formulation, which