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We construct and analyze a group of immersed finite element (IFE) spacesformed by linear, bilinear and rotated Q1 polynomials for solving planarelasticity equation involving interface. The shape functions in these IFEspaces are constructed through a group of approximate jump conditions such thatthe unisolvence of the bilinear and rotated Q1 IFE shape functions are alwaysguaranteed regardless of the Lam\`e parameters and the interface location. Theboundedness property and a group of identities of the proposed IFE shapefunctions are established. A multi-point Taylor expansion is utilized to showthe optimal approximation capabilities for the proposed IFE spaces through theLagrange type interpolation operators.

Approximation Capabilities of Immersed Finite Element Spaces for Elasticity Interface Problems