Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data
Bikram Bir, Deepjyoti Goswami, Amiya K. Pani
Abstract
Abstract In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the two-dimensional Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty }$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal a priori error estimate in $\textbf {L}^2$-norm is derived for the discrete problem with nonsmooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.
Topics & Concepts
MathematicsDiscretizationUniquenessBackward Euler methodMathematical analysisNorm (philosophy)Bounded functionDirichlet distributionA priori and a posterioriApplied mathematicsFinite element methodBoundary value problemEpistemologyPhilosophyPolitical scienceThermodynamicsLawPhysicsFluid Dynamics and Turbulent FlowsAdvanced Numerical Methods in Computational MathematicsRheology and Fluid Dynamics Studies