A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology

This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure

A Virtual Element Method for a Nonlocal FitzHugh-Nagumo Model of Cardiac Electrophysiology

We present a Virtual Element Method (VEM) for a nonlocal reaction-diffusion system of the cardiac electric field. To this system, we analyze an $H^1(\Omega)$-conforming discretization by means of VEM which can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general $L^p$ compactness criterion. Moreover, we obtain optimal order space-time error estimates in the $L^2$ norm. Finally, we report some numerical tests supporting the theoretical results

Entropy Solution for Anisotropic Reaction-Diffusion-Advection Systems with L1 Data

In this paper, we study the question of existence and uniqueness of entropy solutions for a system of nonlinear partial differential equations with general anisotropic diffusivity and transport effects, supplemented with no-flux boundary conditions, modeling the spread of an epidemic disease through a heterogeneous habitat.In this paper, we study the question of existence and uniqueness of entropy solutions for a system of nonlinear partial differential equations with general anisotropic diffusivity and transport effects, supplemented with no-flux bound ary conditions, modeling the spread of an epidemic disease through a heteroge neous habitat

Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems

We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these equations in these applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree. By a thresholding procedure, namely the elimination of leaves that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen optimally, in the sense that the total error of the adaptive scheme is of the same slope as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.Comment: 27 pages, 14 figure

Well-posedness results for triply nonlinear degenerate parabolic equations

We study the well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problem $$b(u)_t - {\rm div} \tilde{\mathfrak a}(u,\nabla\phi(u))+\psi(u)=f, \quad u|_{t=0}=u_0$$ in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities $b,\phi$ and $\psi$ are supposed to be continuous non-decreasing, and the nonlinearity $\tilde{\mathfrak a}$ falls within the Leray-Lions framework. Some restrictions are imposed on the dependence of $\tilde{\mathfrak a}(u,\nabla\phi(u))$ on $u$ and also on the set where $\phi$ degenerates. A model case is $\tilde{\mathfrak a}(u,\nabla\phi(u)) =\tilde{\mathfrak{f}}(b(u),\psi(u),\phi(u))+k(u)\mathfrak{a}_0(\nabla\phi(u)),$ with $\phi$ which is strictly increasing except on a locally finite number of segments, and $\mathfrak{a}_0$ which is of the Leray-Lions kind. We are interested in existence, uniqueness and stability of entropy solutions. If $b=\mathrm{Id}$, we obtain a general continuous dependence result on data $u_0,f$ and nonlinearities $b,\psi,\phi,\tilde{\mathfrak{a}}$. Similar result is shown for the degenerate elliptic problem which corresponds to the case of $b\equiv 0$ and general non-decreasing surjective $\psi$. Existence, uniqueness and continuous dependence on data $u_0,f$ are shown when $[b+\psi](\R)=\R$ and $\phi\circ [b+\psi]^{-1}$ is continuous

Convergence of discrete duality finite volume schemes for the cardiac bidomain model

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented

Structural stability for variable exponent elliptic problems. II. The p(u)p(u)-laplacian and coupled problems.

International audienceWe study well-posedness for elliptic problems under the form b(u)-\div \mathfrak{a}(x,u,\Grad u)=f, where $\mathfrak{a}$ satisfies the classical Leray-Lions assumptionswith an exponent $p$ that may depend both on the space variable $x$ and on the unknown solution $u$. A prototype case is the equation u-\div \Bigl( |\grad u|^{p(u)-2}\grad u \Bigr)=f. We have to assume that \inf_{x\in\overline{\Om},\,z\in\R} p(x,z) is greater than the space dimension $N$. Then, under mild regularity assumptions on \Om and on the nonlinearities, we show that the associated solution operator is an order-preserving contraction in L^1(\Om). In addition, existence analysis for a sample coupled system for unknowns $(u,v)$ involving the $p(v)$-laplacian of $u$ is carried out. Coupled elliptic systems with similar structure appear in applications, e.g. in modelling of stationary thermo-rheological fluids

Structural stability for variable exponent elliptic problems. I. The p(x)p(x)-laplacian kind problems.

the first version of the preprint is now split into two partsInternational audienceWe study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the elliptic problems under the form b(u_n)-\div mathfrak{a}_n(x,\Grad u_n)=f_n. The equation is set in a bounded domain \Om of $\R^N$ and supplied with the homogeneous Dirichlet boundary condition on \ptl\Om. Here $b$ is a non-decreasing function on $\R$, and $\Bigl(\mathfrak{a}_n(x,\xi)\Bigr)_n$ is a family of applications which verifies the classical Leray-Lions hypotheses but with a variable summability exponent $p_n(x)$, $