arXiv subjects
Publications and source records attributed to B. V. Rathish Kumar.
In this study we derive the Finite element apriori error estimate for the monodomain cardiac electric model in conjunction with the generic form for a class of nonlinear ionic models. The analysis establishes a $o(h^2+k)$ space-time convergence. Further the computational realization of the same for different reduced ionic test models is presented.
In this study, we consider a system of degenerate reaction-diffusion equations, which govern the electric activity in the heart with a diffusion term modeling the potential in the surrounding tissue and the nonlinear ionic model proposed by Morris $\&$ Lecar. The global existence of a solution is established based on regularization argument using Fedo-Galerkin/Compactness approach. The uniqueness of the solution is shown based on Gronwell's Lemma upon some special treatment of nonlinear terms. The system of the continuous space-time model is first reduced to a semi-discrete time-dependent system based on finite element formulation, and then the fully discrete system is derived using the Backward Euler time stepping scheme. The numerical solution obtained using FreeFem++ are presented.
In this study a stabilized finite element method for solving advection-diffusion-reaction equation with spatially variable coefficients has been carried out. Here subgrid scale approach along with algebraic approximation to the sub-scales has been chosen as stabilized method among various other methods. Both a priori and a posteriori finite element error estimates in $L_2$ norms have been derived after introducing the stabilized formulation of the variational form. An expression of the stabilization parameter for this 2D problem has also been derived here. At last numerical experiments are presented to verify numerical performance of the stabilized method and check the credibility of the theoretically derived expression of the stabilization parameter.
We propose a non uniform web spline based finite element analysis for elliptic partial differential equation with the gradient type nonlinearity in their principal coefficients like p-laplacian equation and Quasi-Newtonian fluid flow equations. We discuss the well-posednes of the problems and also derive the apriori error estimates for the proposed finite element analysis and obtain convergence rate of $\mathcal{O}(h^α)$ for $α> 0$.
We analyze the error of the WEB-S finite element method applied to elliptic systems with non-cooperative dominant coupling,with a mixed Dirichlet/Neumann/Robin boundary condition. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. These results are based on an extensive regularity analysis of the interface problems of concern.Finally, the error analysis is illustrated by numerical experiments.
In this study, we propose a new scheme named as complete flux scheme (CFS) based on the finite volume method for solving singularly perturbed differential-difference equations (SPDDEs) of elliptic type. An alternate integral representation for the flux is obtained which plays an important role in the derivation of CF scheme. We have established the stability, consistency and quadrature convergence of the proposed scheme. The scheme is successfully implemented on test problems.
This text proposes a fast, rapidly convergent Nyström method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper.