In this paper we devise a moving mesh finite element method

In this paper we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. in many areas; in particular in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way GSK256066 interaction of a migrating cell in a chemotactic field where the bulk region corresponds to the extracellular region and the surface to the cell membrane. onto the physically evolving domain located at on has velocity so that points on evolve with a velocity field denote the unit outward normal to and let be any open subset of containing which is differentiable in by denotes the usual gradient on is defined as the tangential divergence of the tangential gradient is the concentration at position at time will be coupled to through the generally nonlinear flux boundary condition denotes the concentration of at the point and is the outward unit normal to leading to GSK256066 a flux off of is the boundary diffusion coefficient is a surface reaction term. 2.1 ALE reformulation For the reasons mentioned previous when the site is moving a common frame of research used for computational reasons may be GSK256066 the Arbitrary Lagrangian Eulerian (ALE) frame. Permit be considered a grouped category of bijective mappings which in each with coordinates with coordinates to section 4. For an arbitrary function may be the corresponding function in the ALE framework; that’s as on will generally be different through the materials speed the ALE change will become solely Lagrangian in character. To relate enough time derivatives with regards to the ALE change to the materials derivative a typical software of the string rule Rabbit Polyclonal to TNAP2. provides of test features on the site can be a function described on in a way that with usually do not rely on time after that for any we are able to set up from (11) that and the usage of (3) (12) and (13) provides conservative weak type: find in a way that such that and so are approximated by polygonal domains and it is covered by a set triangulation with right edges in order that can be chosen to become this is the boundary of will become denoted by will become denoted by and the amount of vertices for the boundary as as may be the space of linear polynomials on of the proper execution denotes the positioning of node at period is the connected nodal basis function in become the picture from the research triangulation beneath the GSK256066 discrete ALE mapping which may be the picture of a triangle can be therefore thought as such that in a way that and so are the time-dependent bulk and surface area nodal basis features. If and you will be sparse as just those ideals of related to boundary vertices will be non-zero. The spatial discretisation of the boundary equation (19) results in a system of ODEs are the appropriately reordered nonzero elements of into equal time intervals of size and denote is the piecewise linear map at time using a semi-implicit backward Euler method where the linear diffusion and mesh movement terms are treated implicitly and the nonlinear reaction and coupling terms are treated explicitly. The predicted boundary solution therefore satisfies the linear system and are nonlinear. The linear systems arising above are solved using the iterative method BiCGSTAB [52] and an incomplete LU (ILU) factorisation as a preconditioner. For the cell migration application considered later we note that the diffusive time scales in the extracellular region are often much shorter than the time scale associated with cell migration. As the time integration scheme above is fully implicit in the diffusive terms it is therefore robust to the choice of the time step for these applications. 3.3 A model bulk-surface problem on a stationary domain To get an indication of the spatial and temporal convergence rate of the coupled bulk-surface finite element discretisation we apply it to the solution of the following model problem: so that is the first-order Bessel function of the first kind and and a time step was used. Fig. 3(a) shows the maximum error over all grid nodes for both the bulk and surface numerical solutions and we can see that both converge at the price of elements and different period steps. We are able to discover from Fig. 3(b) the fact that three-step solution treatment (24) (25) (26) furnishes approximations that are second-order accurate with time. Remember that if the top solution correction stage (26) had been omitted then needlessly to say the ensuing approximations were just temporally first-order accurate. Fig. 2 Numerical option of combined model problem.