This paper is devoted to the numerical analysis of abstract parabolic problem <svg style="vertical-align:-2.21957pt;width:80.474998px;" id="M1" height="15.325" version="1.1" viewBox="0 0 80.474998 15.325" width="80.474998"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,15.325)">
		<g transform="translate(72,-59.74)">
			<text transform="matrix(1,0,0,-1,-71.95,62)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-66.02,67.01)">
				<tspan style="font-size: 8.75px; " x="0" y="0"></tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-62.55,62)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑡</tspan>
				<tspan style="font-size: 12.50px; " x="8.0769377" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="15.703768" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="27.744158" y="0">𝐴</tspan>
				<tspan style="font-size: 12.50px; " x="36.708809" y="0">𝑢</tspan>
				<tspan style="font-size: 12.50px; " x="42.635231" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="46.798729" y="0">𝑡</tspan>
				<tspan style="font-size: 12.50px; " x="50.71217" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>;
<svg style="vertical-align:-2.21957pt;width:58.650002px;" id="M2" height="16.5375" version="1.1" viewBox="0 0 58.650002 16.5375" width="58.650002"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,16.5375)">
		<g transform="translate(72,-58.77)">
			<text transform="matrix(1,0,0,-1,-71.95,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan>
				<tspan style="font-size: 12.50px; " x="5.9264221" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="10.089921" y="0">0</tspan>
				<tspan style="font-size: 12.50px; " x="16.341421" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="23.96825" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="36.00864" y="0">𝑢</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-30.01,66.03)">
				<tspan style="font-size: 8.75px; " x="0" y="0">0</tspan>
			</text>

		</g>
	</g>
</svg>, with hyperbolic generator <svg style="vertical-align:-0.0pt;width:11.325px;" id="M3" height="10.55" version="1.1" viewBox="0 0 11.325 10.55" width="11.325"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.55)">
		<g transform="translate(72,-63.56)">
			<text transform="matrix(1,0,0,-1,-71.95,63.61)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝐴</tspan>
			</text>

		</g>
	</g>
</svg>. We are developing a general approach to establish
a discrete dichotomy in a very general setting in case of discrete approximation in space and
time. It is a well-known fact that the phase space in the neighborhood of the
hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced
to initial value problems with exponential decaying solutions in opposite time direction. We use
the theory of compact approximation principle and collectively condensing approximation to
show that such a decomposition of the flow persists under rather general approximation schemes. 
The main assumption of our results is naturally satisfied, in particular, for operators with
compact resolvents and condensing semigroups and can be verified for finite element as well as
finite difference methods.

Javier Pastor

Sergey Piskarev

The Exponential Dichotomy under Discretization on General Approximation Scheme

The aim of this paper is to study a nonlinear stationary Stokes problem with mixed boundary conditions that describes the ice velocity and pressure fields of grounded glaciers under Glen's flow law. Using convex analysis arguments, we prove the existence and the uniqueness of a weak solution. A finite element method is applied with approximation spaces that satisfy the inf-sup condition, and <i>a priori</i> error estimates are established by using a quasinorm technique. Several algorithms (including Newton's method) are proposed to solve the
nonlinearity of the Stokes problem and are proved to be convergent. Our results are supported by numerical convergence studies.

Guillaume Jouvet

Jacques Rappaz

Analysis and Finite Element Approximation of a Nonlinear Stationary Stokes Problem Arising in Glaciology

We present an incomplete UL (IUL) decomposition of matrix <svg style="vertical-align:-0.0pt;width:11.325px;" id="M1" height="10.55" version="1.1" viewBox="0 0 11.325 10.55" width="11.325"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.55)">
		<g transform="translate(72,-63.56)">
			<text transform="matrix(1,0,0,-1,-71.95,63.61)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝐴</tspan>
			</text>

		</g>
	</g>
</svg> which is extracted as a by-product of BFAPINV (backward factored approximate inverse) process. We term this IUL factorization as IULBF. We have used ILUFF [3] and IULBF as left preconditioner for linear systems. Different versions of ILUFF and IULBF preconditioners are computed by  using  different dropping techniques. In this paper, we compare quality of different versions of ILUFF and IULBF preconditioners.

Amin Rafiei

Fatemeh Shahlaei

Different Versions of ILU and IUL Factorizations Obtained from Forward and Backward Factored Approximate Inverse Processes&#x2014;Part I

This paper contributes a very general class of two-point iterative methods without memory for solving nonlinear equations. The class of methods is developed using weight function approach. Per iteration, each method of the class includes two evaluations of the function and one of its first-order derivative. The analytical study of the main theorem is presented in detail to show the fourth order of convergence. Furthermore, it is discussed that many of the existing fourth-order methods without memory are members from this developed class. Finally, numerical examples are taken into account to manifest the accuracy of the derived methods.

F. Soleymani

Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations

The aim of this paper is to present the structure of a class of
matrices that enables explicit inverse to be obtained. Starting from an
already known class of matrices, we construct a Hadamard product that
derives the class under consideration. The latter are defined by <svg style="vertical-align:-0.10033pt;width:41.174999px;" id="M1" height="10.8125" version="1.1" viewBox="0 0 41.174999 10.8125" width="41.174999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.8125)">
		<g transform="translate(72,-63.35)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">4</tspan>
				<tspan style="font-size: 12.50px; " x="6.2515001" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="15.241157" y="0">−</tspan>
				<tspan style="font-size: 12.50px; " x="26.581379" y="0">2</tspan>
			</text>

		</g>
	</g>
</svg>
parameters, analytic expressions of which provide the elements of the
lower Hessenberg form inverse. Recursion formulae of these expressions
reduce the arithmetic operations in evaluating the inverse to <svg style="vertical-align:-2.21957pt;width:35.212502px;" id="M2" height="16.5375" version="1.1" viewBox="0 0 35.212502 16.5375" width="35.212502"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,16.5375)">
		<g transform="translate(72,-58.77)">
			<text transform="matrix(1,0,0,-1,-71.95,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝒪</tspan>
				<tspan style="font-size: 12.50px; " x="8.6520758" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="12.815575" y="0">𝑛</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-52.92,66.03)">
				<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-48.05,61.03)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>.

F. N. Valvi

A Generalization of a Class of Matrices: Analytic Inverse and Determinant

We present new iterative methods based on refinement process for solving large sparse Sylvester-observer
equations applied in state estimation of a continuous-time system. These methods use projection methods to produce low-dimensional Sylvester-observer matrix equations that are solved by the direct methods. Moreover, the refinement process
described in this paper has the capability of improving the results obtained by any other methods. Some numerical results
will be reported to illustrate the efficiency of the proposed methods.

H. Saberi Najafi

A. H. Refahi Sheikhani

Refinement Methods for State Estimation via Sylvester-Observer Equation

A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form <svg style="vertical-align:-3.25793pt;width:232.91251px;" id="M1" height="16.612499" version="1.1" viewBox="0 0 232.91251 16.612499" width="232.91251"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,16.6125)">
		<g transform="translate(72,-58.71)">
			<text transform="matrix(1,0,0,-1,-71.95,62)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑦</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-65.75,67.01)">
				<tspan style="font-size: 8.75px; " x="0" y="0"></tspan>
				<tspan style="font-size: 8.75px; " x="2.9756801" y="0"></tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-55.83,62)">
				<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="12.040389" y="0">𝑓</tspan>
				<tspan style="font-size: 12.50px; " x="18.979553" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="23.143053" y="0">𝑥</tspan>
				<tspan style="font-size: 12.50px; " x="30.019703" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="35.220951" y="0">𝑦</tspan>
				<tspan style="font-size: 12.50px; " x="41.42244" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="46.636189" y="0">𝑦</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-2.99,67.01)">
				<tspan style="font-size: 8.75px; " x="0" y="0"></tspan>
			</text>

			<text transform="matrix(1,0,0,-1,0.49,62)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="9.364747" y="0">𝑦</tspan>
				<tspan style="font-size: 12.50px; " x="15.566235" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="19.729734" y="0">𝑥</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,27.1,58.88)">
				<tspan style="font-size: 8.75px; " x="0" y="0">0</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,31.98,62)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="7.6268301" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="19.667219" y="0">𝑎</tspan>
				<tspan style="font-size: 12.50px; " x="25.943726" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="31.144974" y="0">𝑦</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,69.34,67.01)">
				<tspan style="font-size: 8.75px; " x="0" y="0"></tspan>
			</text>

			<text transform="matrix(1,0,0,-1,72.81,62)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑥</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,83.85,58.88)">
				<tspan style="font-size: 8.75px; " x="0" y="0">0</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,88.73,62)">
				<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
				<tspan style="font-size: 12.50px; " x="7.6268301" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="19.667219" y="0">𝑏</tspan>
			</text>

		</g>
	</g>
</svg>. The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy.

J. O. Ehigie

S. A. Okunuga

A. B. Sofoluwe

3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations

A new Monte Carlo approach for evaluating the generalized eigenpair
of real symmetric matrices will be proposed. Algorithm for the three smallest eigenpairs
based on the partitioning inverse Monte Carlo iterative (IMCI) method will be considered.

Behrouz Fathi Vajargah

Farshid Mehrdoust

Partitioning Inverse Monte Carlo Iterative Algorithm for Finding the Three Smallest Eigenpairs of Generalized Eigenvalue Problem

The unsteady, buoyancy-induced, hydromagnetic, thermal convection flow in a semi-infinite porous regime adjacent to an infinite hot vertical plate moving with constant velocity, is studied in the presence of significant thermal radiation. The momentum and energy conservation equations are normalized and then solved using both the Laplace transform technique and Network Numerical Simulation. Excellent agreement is obtained between both analytical and numerical methods. An increase in  Hartmann number (<svg style="vertical-align:-0.0pt;width:20.987499px;" id="M1" height="12.3625" version="1.1" viewBox="0 0 20.987499 12.3625" width="20.987499" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,12.3625)"><g transform="translate(72,-62.11)"><text transform="matrix(1,0,0,-1,-71.95,62.16)"><tspan style="font-size: 11.21px; " x="0" y="0">𝑀</tspan></text><text transform="matrix(1,0,0,-1,-59.68,66.65)"><tspan style="font-size: 7.85px; " x="0" y="0">2</tspan></text></g></g></svg>) strongly decelerates the flow and for very high strength magnetic fields <svg style="vertical-align:-1.99126pt;width:61.712502px;" id="M2" height="14.8375" version="1.1" viewBox="0 0 61.712502 14.8375" width="61.712502" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,14.8375)"><g transform="translate(72,-60.13)"><text transform="matrix(1,0,0,-1,-71.95,62.16)"><tspan style="font-size: 11.21px; " x="0" y="0">(</tspan><tspan style="font-size: 11.21px; " x="3.732264" y="0">𝑀</tspan></text><text transform="matrix(1,0,0,-1,-55.95,66.65)"><tspan style="font-size: 7.85px; " x="0" y="0">2</tspan></text><text transform="matrix(1,0,0,-1,-48.41,62.16)"><tspan style="font-size: 11.21px; " x="0" y="0">=</tspan><tspan style="font-size: 11.21px; " x="10.782096" y="0">2</tspan><tspan style="font-size: 11.21px; " x="16.386095" y="0">0</tspan><tspan style="font-size: 11.21px; " x="21.990095" y="0">)</tspan></text></g></g></svg>, the flow is reversed after a short time interval. The classical velocity overshoot is also detected close to the plate surface for low to intermediate values of <svg style="vertical-align:-0.0pt;width:20.987499px;" id="M3" height="12.3625" version="1.1" viewBox="0 0 20.987499 12.3625" width="20.987499" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,12.3625)"><g transform="translate(72,-62.11)"><text transform="matrix(1,0,0,-1,-71.95,62.16)"><tspan style="font-size: 11.21px; " x="0" y="0">𝑀</tspan></text><text transform="matrix(1,0,0,-1,-59.68,66.65)"><tspan style="font-size: 7.85px; " x="0" y="0">2</tspan></text></g></g></svg> at both small and large times;  however this overshoot vanishes for larger strengths of the transverse magnetic  field <svg style="vertical-align:-1.99126pt;width:61.712502px;" id="M4" height="14.8375" version="1.1" viewBox="0 0 61.712502 14.8375" width="61.712502" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,14.8375)"><g transform="translate(72,-60.13)"><text transform="matrix(1,0,0,-1,-71.95,62.16)"><tspan style="font-size: 11.21px; " x="0" y="0">(</tspan><tspan style="font-size: 11.21px; " x="3.732264" y="0">𝑀</tspan></text><text transform="matrix(1,0,0,-1,-55.95,66.65)"><tspan style="font-size: 7.85px; " x="0" y="0">2</tspan></text><text transform="matrix(1,0,0,-1,-48.41,62.16)"><tspan style="font-size: 11.21px; " x="0" y="0">=</tspan><tspan style="font-size: 11.21px; " x="10.782096" y="0">1</tspan><tspan style="font-size: 11.21px; " x="16.386095" y="0">0</tspan><tspan style="font-size: 11.21px; " x="21.990095" y="0">)</tspan></text></g></g></svg>.  An increase in radiation-conduction parameter (<svg style="vertical-align:-2.81255pt;width:15.425px;" id="M5" height="12.775" version="1.1" viewBox="0 0 15.425 12.775" width="15.425" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,12.775)"><g transform="translate(72,-61.78)"><text transform="matrix(1,0,0,-1,-71.95,64.63)"><tspan style="font-size: 11.21px; " x="0" y="0">𝐾</tspan></text><text transform="matrix(1,0,0,-1,-63.41,61.83)"><tspan style="font-size: 7.85px; " x="0" y="0">𝑟</tspan></text></g></g></svg>) significantly increases temperature  throughout the porous regime at both small and larger times, adjacent to the plate, but decreases the shear stress magnitudes at the plate. Temperature gradient is reduced at the plate surface for all times, with a rise in radiation-conduction parameter (<svg style="vertical-align:-2.81255pt;width:15.425px;" id="M6" height="12.775" version="1.1" viewBox="0 0 15.425 12.775" width="15.425" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,12.775)"><g transform="translate(72,-61.78)"><text transform="matrix(1,0,0,-1,-71.95,64.63)"><tspan style="font-size: 11.21px; " x="0" y="0">𝐾</tspan></text><text transform="matrix(1,0,0,-1,-63.41,61.83)"><tspan style="font-size: 7.85px; " x="0" y="0">𝑟</tspan></text></g></g></svg>). Shear stress is reduced considerably with an increase in Darcian drag parameter (<svg style="vertical-align:-4.25366pt;width:16.2125px;" id="M7" height="14.575" version="1.1" viewBox="0 0 16.2125 14.575" width="16.2125" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(1.25,0,0,-1.25,0,14.575)"><g transform="translate(72,-60.34)"><text transform="matrix(1,0,0,-1,-71.95,64.63)"><tspan style="font-size: 11.21px; " x="0" y="0">𝐾</tspan></text><text transform="matrix(1,0,0,-1,-63.41,61.83)"><tspan style="font-size: 7.85px; " x="0" y="0">𝑝</tspan></text></g></g></svg>).

O. Anwar B&#xe9;g

J. Zueco

S. K. Ghosh

Alireza Heidari

Unsteady Magnetohydrodynamic Heat Transfer in a Semi-Infinite Porous Medium with Thermal Radiation Flux: Analytical and Numerical Study

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Advances in Numerical Analysis