### Solution of the Non-Linear Parabolic Problems Using Nonlinear Complementarity Algorithm (FDA-NCP)

#### Abstract

Parabolic type problems involving a variational and complementarity formulation arise in mathematical models of several applications in Engineering, Economy, Biology and different branches of Physics. These kinds of problems present several analytical and numerical difﬁculties related, for example, to time evolution and moving boundary.

In this work we implement a numerical method based on the ﬁnite difference scheme for time evolution and nonlinear complementarity algorithm (FDA-NCP) for solving the problem at each time step. We use the implicit ﬁnite difference scheme with adaptative time step implementation which allows us to use bigger time steps and speed up the simulations. One of the advantages using the FDA-NCP is its global convergence.

This method was applied to simple non-linear parabolic partial differential equation, which describes oxygen diffusion problem inside one cell. This equation was rewritten in the quasi-variational form. The main problem consists in tracking the moving boundary that represents the oxygen penetration depth inside the cell.

In this work we implement a numerical method based on the ﬁnite difference scheme for time evolution and nonlinear complementarity algorithm (FDA-NCP) for solving the problem at each time step. We use the implicit ﬁnite difference scheme with adaptative time step implementation which allows us to use bigger time steps and speed up the simulations. One of the advantages using the FDA-NCP is its global convergence.

This method was applied to simple non-linear parabolic partial differential equation, which describes oxygen diffusion problem inside one cell. This equation was rewritten in the quasi-variational form. The main problem consists in tracking the moving boundary that represents the oxygen penetration depth inside the cell.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**