The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains

This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used

The inf-sup constant for the divergence on corner domains

The inf-sup constant for the divergence, or LBB constant, is related to the Cosserat spectrum. It has been known for a long time that on non-smooth domains the Cosserat operator has a non-trivial essential spectrum, which can be used to bound the LBB constant from above. We prove that the essential spectrum on a plane polygon consists of an interval related to the corner angles and that on three-dimensional domains with edges, the essential spectrum contains such an interval. We obtain some numerical evidence for the extent of the essential spectrum on domains with axisymmetric conical points by computing the roots of explicitly given holomorphic functions related to the corner Mellin symbol. Using finite element discretizations of the Stokes problem, we present numerical results pertaining to the question of the existence of eigenvalues below the essential spectrum on rectangles and cuboids

A Mixed Method for Axisymmetric Div-Curl Systems

We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties

Stabilized mixed approximation of axisymmetric Brinkman flows

This paper is devoted to the numerical analysis of an augmented finite element approximation of the axisymmetric Brinkman equations. Stabilization of the variational formulation is achieved by adding suitable Galerkin least-squares terms, allowing us to transform the original problem into a formulation better suited for performing its stability analysis. The sought quantities (here velocity, vorticity, and pressure) are approximated by Raviart−Thomas elements of arbitrary order k ≥ 0, piecewise continuous polynomials of degree k + 1, and piecewise polynomials of degree k, respectively. The well-posedness of the resulting continuous and discrete variational problems is rigorously derived by virtue of the classical Babuška–Brezzi theory. We further establish a priori error estimates in the natural norms, and we provide a few numerical tests illustrating the behavior of the proposed augmented scheme and confirming our theoretical findings regarding optimal convergence of the approximate solutions

Solving Numerically the Static Maxwell Equations in an Axisymmetric Singular Geometry

We propose a new numerical method to compute the singular solution of the Maxwell equations in axisymmetric domains, as for example in non convex polygonal domains. As geometrical singularities are mainly related to the space dependent part of the model, we focus on the static field computation. We then introduce a new approach, that consists in decomposing the domain into two or more subdomains, and to derive an ad hoc variational formulation in each subdomain. The interface conditions are then imposed with a method deduced from a Nitsche method coupled with a specific “exchange” approach. An advantage of this domain decomposition method is that it does not require neither overlapping nor iteration process. Another advantage is that no particular mesh refinement is needed near the geometrical singularities. Numerical examples will be shown

A finite element method for a fourth order surface equation with application to the onset of cell blebbing

A variational problem for a fourth order parabolic surface partial differential equation is discussed. It contains nonlinear lower order terms, on which we only make abstract assumptions, and which need to be defined for specified problems.We derive a semi-discrete scheme based on the surface finite element method, show a-priori error estimates, and use the analytical results to prove well-posedness. Furthermore, we present a computational framework where specific problems can be conveniently implemented and, later on, altered with relative ease. It uses a domain specific language implemented in Python. The high level program control can also be done within the Python scripting environment. The computationally expensive step of evolving the solution over time is carried out by binding to an efficient C++ software back-end. The study is motivated by cell blebbing, which can be instrumental for cell migration. Starting with a force balance for the cell membrane, we derive a continuum model for some mechanical and geometrical aspects of the onset of blebbing in a form that fits into the abstract framework. It is flexible in that it allows for amending force contributions related to membrane tension or the presence of linker molecules between membrane and cell cortex. Cell membrane geometries given in terms of a parametrisation or obtained from image data can be accounted for by the software. The use of a domain specific language to describe the model makes is straightforward to add additional effects such as reaction-diffusion equations modelling some biochemistry on the cell membrane.Some numerical simulations illustrate the approach