Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs

This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates.

ORDERED ANALYTIC REPRESENTATION OF PDES BY HAMILTONIAN CANONICAL SYSTEM

Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessary and sufficient conditions are obtained

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

The present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method (GM ), Galerkin method with weak form (GM / WF ), Petrov-Galerkin method ( PGM), weighted residual method (WRY ), and least squares method or process ( LSM or LSP ) to construct finite element approximations in time. A correspondence is established between these integral forms and the elements of the calculus of variations: 1) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) computational processes for which types of operators and, 2) to establish which integral forms do not yield unconditionally stable computations (variationally inconsistent integral forms, VIC ). It is shown that variationally consistent time integral forms in hpk framework yield computational processes for ODEs in time that are unconditionally stable, provide a mechanism of higher order global differentiability approximations as well as higher degree local approximations in time, provide control over approximation error when used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical studies are presented using standard model problems from the literature and the results are compared with Wilson’s θ method as well as Newmark method to demonstrate highly meritorious features of the proposed methodology.

The Principal Eigenvalue for Jump Processes

A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is presented.The result is complete in the sense that the condition is fulfilled and the resulting bound is sharp for Markov chains under some mild assumptions.

Finite element model to study the effect of exogenous buffer on calcium dynamics in dendritic spines

Dendritic spine plays an important role in calcium regulation in a neuron cell.It serves as a storage site for synaptic strength and receives input from a single synapse of axon.In order to understand the calcium dynamics in a neuron cell,it is crucial to understand the calcium dynamics in dendritic spines.In this paper,an attempt has been made to study the calcium dynamics due to the exogenous buffers,in dendritic spines with the help of a sectional model.The compartments of dendritic spines are discretized using triangular elements.Appropriate boundary conditions have been framed.Finite element method has been employed to obtain the solution in the region for a two-dimensional unsteady state case.MATLAB 7.11 is used for simulation of the problem and numerical computations.The numerical results have been used to study the effect of exogenous buffers on calcium distribution in dendritic spines.

Modeling the mechanics of calcium regulation in T lymphocyte:A finite element method approach

Changes in cellular Ca2+concentration control a variety of physiological activities including hormone and neurotransmitter release,muscular contraction,synaptic plas-ticity,ionic channel permeability,apoptosis,enzyme activity,gene transcription and reproduction process.Spatial-temporal Ca2+dynamics due to Ca2 t release,buffering and re-uptaking plays a central role in studying the Ca2+regulation in T lympho-cytes.In most cases,Ca2+has its major signaling function when it is elevated in the cytosolic compartment.In this paper,a two-dimensional mathematical model to study spatiotemporal variations of intracellular Ca2+concentration in T lymphocyte cell is proposed and investigated.The cell is assumed to be a circular shaped geomnetrical domain for the representation of properties of Ca2+dynamics within the cell includ-ing important parameters.Ca2+binding proteins for the dynamics of Ca2+are itself buffer and other physiological parameters located in Ca2+stores.The model incorpo-rates the important biophysical processes like difusion,reaction,voltage gated Ca2+channel,leak from endoplasmic reticulum(ER),efflux from cytosol to ER via sarco ER Ca2+adenosine triphosphate(SERCA)pumps,buffers and Na+/Ca2+exchanger.The proposed mathematical model is solved using a finite difference method and the finite element method.Appropriate initial and boundary conditions are incorporated in the model based on biophysical conditions of the problem.Computer simulations in MAT-LAB R2019b are employed to investigate mathematical models of reaction-diffusion equation.The effect of source,buffer,Nat/Ca2+exchanger,etc.on spatial and tempo-ral patterns of Ca2+in T lymphocyte has been studied with the help of numerical results.From the obtained results,it is observed that,the coordinated combination of the incor-porated parameters plays a significant role in Ca2+regulation in T lymphocytes.ER leak and voltage-gated Ca2+channel provides the necessary Ca2+to the cell when required for its proper functioning,while on the other side buffers,SERCA pump and Na+/Ca2+exchanger makes balance in the Ca2+concentration,so as to prevent the cell from death as higher concentration for longer time is harmful for the cell and can cause cell death.