The purpose of this paper is to introduce a level set framework suitable for identifying heart infarctions. We do this by introducing a modified Monodomain model, and an output least squares formulation comparing simulation results with observation data. In this way we obtain a flexible methodology allowing us to approximately determine the characteristics of infarctions with rather complex geometrical properties. Our approach involves a CPU demanding minimization problem. This problem is solved by applying an adjoint equation enabling efficient differentiation of the involved cost-functional. Finally, our theoretical findings are illuminated by a series of numerical experiments.

We investigate an explicit finite difference scheme for a Beeler-Reuter based model of cardiac electrical activity. As our main result, we prove that the finite difference solutions are bounded in the $L^∞$-norm. We also prove the existence of a weak solution by showing convergence to the solutions of the underlying model as the discretization parameters tend to zero. The convergence proof is based on the compactness method.

In this paper, we present what we believe is the first a posteriori finite element error analysis for the system of equations that governs micromachined microsensors. Our main result is the establishment of an efficient and reliable a posteriori error estimator $\Theta$.

We consider an optimal control problem described by semilinear parabolic partial differential equations, with control and state constraints. Since this problem may have no classical solutions, it is also formulated in the relaxed form. The classical control problem is then discretized by using a finite element method in space and the implicit Crank-Nicolson midpoint scheme in time, while the controls are approximated by classical controls that are bilinear on pairs of blocks. We prove that strong accumulation points in $L^2$ of sequences of optimal (resp. admissible and extremal) discrete controls are optimal (resp. admissible and weakly extremal classical) for the continuous classical problem, and that relaxed accumulation points of sequences of optimal (resp. admissible and extremal relaxed) discrete controls are optimal (resp. admissible and weakly extremal relaxed) for the continuous relaxed problem. We then apply a penalized gradient projection method to each discrete problem, and also a progressively refining version of the discrete method to the continuous classical problem. Under appropriate assumptions, we prove that accumulation points of sequences generated by the first method are admissible and extremal for the discrete problem, and that strong classical (resp. relaxed) accumulation points of sequences of discrete controls generated by the second method are admissible and weakly extremal classical (resp. relaxed) for the continuous classical (resp. relaxed) problem. For nonconvex problems whose solutions are non-classical, we show that we can apply the above methods to the problem formulated in Gamkrelidze relaxed form. Finally, numerical examples are given.

An axiomatic approach to the numerical approximation $Y$ of some stochastic process $X$ with values on a separable Hilbert space $H$ is presented by means of Lyapunov-type control functions $V$. The processes $X$ and $Y$ are interpreted as flows of stochastic differential and difference equations, respectively. The main result is the proof of some extensions of well-known deterministic principle of Kantorovich-Lax-Richtmeyer to approximate solutions of initial value differential problems to the stochastic case. The concepts of invariance, smoothness of martingale parts, consistency, stability, and contractivity of stochastic processes are uniquely combined to derive efficient convergence rates on finite and infinite time-intervals. The applicability of our results is explained with drift-implicit backward Euler methods applied to ordinary stochastic differential equations (SDEs) driven by standard Wiener processes on Euclidean spaces $H = \mathbb{R}^d$ along functions such as $V(x) = \sum_{i=0} ^k c_i x^{2i}$. A detailed discussion on an example with cubic nonlinearity from field theory in physics (stochastic Ginzburg-Landau equation) illustrates the suggested axiomatic approach.

Penetration of chloride ions into concrete and diffusion of moisture in concrete are important factors responsible for the corrosion of steel in concrete. The two diffusion processes are coupled. This paper deals with the analysis and simulation of coupled chloride penetration and moisture diffusion in concrete. Of particular interest is the parallel programming in finite element method for solving the coupled diffusion problem. Parallel computing technology has been advantageous for solving computationally intensive problems. It has become quite mature technology and more affordable for general application. Our approach to solve the parallel programming problem is to use available libraries, i.e. Portable, Extensible Toolkit for Scientific Computation (PETSc) and The Message Passing Interface (MPI) standard. The formulation of the coupled diffusion problem, the material models involved in the differential equations, the details of parallel domain decomposition technique in the finite element algorithm are presented. The advantages of parallel programming are demonstrated by a numerical example.

This paper studies error estimations for a fully discrete, single step finite element scheme for linear parabolic partial differential equations. Convergence in the norm of the solution space is shown and various error estimates in this norm are derived. In contrast to like results in the extant literature, the error estimates are derived in a stronger norm and under minimal regularity assumptions.