## A convergence analysis of an $$h$$-version finite element method with high-order elements for two-dimensional elasto-plasticity problems.(English)Zbl 0879.73070

Summary: We give an $$h$$-version finite element method for a two-dimensional nonlinear elasto-plasticity problem. A family of admissible constitutive laws based on the so-called gauge-function method is introduced first, and then a high-order $$h$$-version semidiscretization scheme is presented. The existence and uniqueness of the solution for the semidiscrete problem are guaranteed by using some special properties of the constitutive law, and finally we will show that as the maximum element size $$h\to 0$$, the solution of the semidiscrete problem will converge to the solution of the continuous problem. The high-order $$h$$-version discretization scheme introduced here is unusual. If the partition of the spatial space only has rectangles or parallelograms involved, then there would not be any limit on the element degree. However, if the partition of the spatial space has some triangular elements, then only certain combinations of finite element spaces for displacement and stress functions can be used. The discretization scheme also provides a useful idea for applications of $$hp$$-version or high-order $$h$$-version finite element methods for two-dimensional problems where the elasto-plastic body is not a polygon, such as a disk or an annulus.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74C99 Plastic materials, materials of stress-rate and internal-variable type 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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