Page 652 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 652
NUMERICAL ANALYSIS AND SCIENTIFIC COMPUTING
A singularly perturbed problem on a Duran-Lombardi mesh
Mirjana Brdar, mirjana.brdar@uns.ac.rs
University of Novi Sad, Serbia
Coauthors: Ljiljana Teofanov, Goran Radojev
We consider a singularly perturbed convection-diffusion problem on unite square whose solu-
tion may have exponential and parabolic boundary layers. The problem is solved numerically by
a finite element method with piecewise bilinear elements on a graded Duran-Lombardi mesh.
We prove uniform convergence of this method in an energy norm. Furthermore, by using a
streamline-diffusion version of the method (SDFEM) we are able to perform analysis of the
supercloseness property of the SDFEM in the corresponding streamline-diffusion norm. Our
analysis offers a choice of parameters which improves stability.
A numerical algorithm for solving problem for a system of essentially
loaded differential equations
Zhazira Kadirbayeva, zhkadirbayeva@gmail.com
Institute of mathematics and mathematical modeling, Kazakhstan
In the present paper we consider the following linear boundary value problem for essentially
loaded differential equations with multi-point conditions:
dx m m+1
= A(t)x + Mj(t)x˙ (θj) + Ki(t)x(θi) + f (t), t ∈ (0, T ), (1)
dt
j=1 i=0
m+1 d ∈ Rn, x ∈ Rn, (2)
Cix(θi) = d,
i=0
where the (n × n) -matrices A(t), Mj(t) (j = 1, m), Ki(t) (i = 0, m + 1), and n-vector-
function f (t) are continuous on [0, T ], Ci (i = 0, m + 1) are constant (n × n) - matrices, and
0 = θ0 < θ1 < θ2 < . . . < θm−1 < θm < θm+1 = T ; x = max |xi|.
i=1,n
Let C([0, T ], Rn) denote the space of continuous functions x : [0, T ] → Rn with the norm
||x||1 = max ||x(t)||.
t∈[0,T ]
A solution to problem (1), (2) is a continuously differentiable on (0, T ) function x(t) ∈
C([0, T ], Rn) satisfying the essentially loaded differential equations (1) and the multi-point
condition (2).
We offer algorithm for solving to linear multi-point boundary value problem for essentially
loaded differential equations (1), (2). Using the properties of essentially loaded differential
equation and assuming the invertibility of the matrix compiled through the coefficients at the
values of the derivative of the desired function at load points, we reduce the considered problem
to a multi-point boundary value problem for loaded differential equations. The parameteriza-
tion method [1] is used for solving this problem. The linear boundary value problem for loaded
differential equations is reduced to equivalent problem consisting the Cauchy problems for sys-
tem of ordinary differential equations with parameters in subintervals, multi-point condition
and continuity conditions. At fixed values of parameters the Cauchy problem for system of
ordinary differential equations in subinterval has a unique solution. This solution is represented
650
A singularly perturbed problem on a Duran-Lombardi mesh
Mirjana Brdar, mirjana.brdar@uns.ac.rs
University of Novi Sad, Serbia
Coauthors: Ljiljana Teofanov, Goran Radojev
We consider a singularly perturbed convection-diffusion problem on unite square whose solu-
tion may have exponential and parabolic boundary layers. The problem is solved numerically by
a finite element method with piecewise bilinear elements on a graded Duran-Lombardi mesh.
We prove uniform convergence of this method in an energy norm. Furthermore, by using a
streamline-diffusion version of the method (SDFEM) we are able to perform analysis of the
supercloseness property of the SDFEM in the corresponding streamline-diffusion norm. Our
analysis offers a choice of parameters which improves stability.
A numerical algorithm for solving problem for a system of essentially
loaded differential equations
Zhazira Kadirbayeva, zhkadirbayeva@gmail.com
Institute of mathematics and mathematical modeling, Kazakhstan
In the present paper we consider the following linear boundary value problem for essentially
loaded differential equations with multi-point conditions:
dx m m+1
= A(t)x + Mj(t)x˙ (θj) + Ki(t)x(θi) + f (t), t ∈ (0, T ), (1)
dt
j=1 i=0
m+1 d ∈ Rn, x ∈ Rn, (2)
Cix(θi) = d,
i=0
where the (n × n) -matrices A(t), Mj(t) (j = 1, m), Ki(t) (i = 0, m + 1), and n-vector-
function f (t) are continuous on [0, T ], Ci (i = 0, m + 1) are constant (n × n) - matrices, and
0 = θ0 < θ1 < θ2 < . . . < θm−1 < θm < θm+1 = T ; x = max |xi|.
i=1,n
Let C([0, T ], Rn) denote the space of continuous functions x : [0, T ] → Rn with the norm
||x||1 = max ||x(t)||.
t∈[0,T ]
A solution to problem (1), (2) is a continuously differentiable on (0, T ) function x(t) ∈
C([0, T ], Rn) satisfying the essentially loaded differential equations (1) and the multi-point
condition (2).
We offer algorithm for solving to linear multi-point boundary value problem for essentially
loaded differential equations (1), (2). Using the properties of essentially loaded differential
equation and assuming the invertibility of the matrix compiled through the coefficients at the
values of the derivative of the desired function at load points, we reduce the considered problem
to a multi-point boundary value problem for loaded differential equations. The parameteriza-
tion method [1] is used for solving this problem. The linear boundary value problem for loaded
differential equations is reduced to equivalent problem consisting the Cauchy problems for sys-
tem of ordinary differential equations with parameters in subintervals, multi-point condition
and continuity conditions. At fixed values of parameters the Cauchy problem for system of
ordinary differential equations in subinterval has a unique solution. This solution is represented
650
