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Mathematics Colloquium

UT El Paso

Title: A weak Galerkin method for linear elasticity
Date: Friday, April 26, 2019
Place and Time: Room 101, Love Building, 3:35-4:25 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. In this talk, I will discuss weak Galerkin (WG) methods for solving partial differential equations, with particular attention to the equations of linear elasticity. WG method is a newly developed numerical technique for solving PDEs where classical differential operators are replaced by discrete weak differential operators in the variational form of the underlying PDE problem.

It is well known that standard continuous Galerkin methods, such as continuous piecewise linear or bilinear elements, yield poor approximations to the displacement as the material becomes incompressible or equivalently, the Lame constant tends to infinity. This phenomenon is known as ``Poisson locking'' and overcoming locking has been the subject of extensive research over several decades.

I will propose a locking-free and lowest-order WG method for the equations of linear elasticity based on the displacement formulation. Unlike other previously proposed WG methods for linear elasticity, the new method does not require a stabilization term for the existence or uniqueness of the solution. I will present a-priori error estimates of optimal order in the discrete H1 and L2 norms for the displacement when the solution is smooth. They are independent of the Lame constant, therefore the performance of the new method does not deteriorate as the material becomes incompressible. Finally, I will present some numerical results to confirm the optimal order error estimates and also to show the locking-free nature of the new method.