An $hp$-adaptive discontinuous Galerkin discretization of a static anti-plane shear crack model
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916464238264320 |
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| author | Manohar, Ram Mallikarjunaiah, S. M. |
| author_facet | Manohar, Ram Mallikarjunaiah, S. M. |
| contents | We propose an $hp$-adaptive discontinuous Galerkin finite element method (DGFEM) to approximate the solution of a static crack boundary value problem. The mathematical model describes the behavior of a geometrically linear strain-limiting elastic body. The compatibility condition for the physical variables, along with a specific algebraically nonlinear constitutive relationship, leads to a second-order quasi-linear elliptic boundary value problem. We demonstrate the existence of a unique discrete solution using Ritz representation theory across the entire range of modeling parameters. Additionally, we derive a priori error estimates for the DGFEM, which are computable and, importantly, expressed in terms of natural energy and $L^2$-norms. Numerical examples showcase the performance of the proposed method in the context of a manufactured solution and a non-convex domain containing an edge crack. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00021 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An $hp$-adaptive discontinuous Galerkin discretization of a static anti-plane shear crack model Manohar, Ram Mallikarjunaiah, S. M. Numerical Analysis We propose an $hp$-adaptive discontinuous Galerkin finite element method (DGFEM) to approximate the solution of a static crack boundary value problem. The mathematical model describes the behavior of a geometrically linear strain-limiting elastic body. The compatibility condition for the physical variables, along with a specific algebraically nonlinear constitutive relationship, leads to a second-order quasi-linear elliptic boundary value problem. We demonstrate the existence of a unique discrete solution using Ritz representation theory across the entire range of modeling parameters. Additionally, we derive a priori error estimates for the DGFEM, which are computable and, importantly, expressed in terms of natural energy and $L^2$-norms. Numerical examples showcase the performance of the proposed method in the context of a manufactured solution and a non-convex domain containing an edge crack. |
| title | An $hp$-adaptive discontinuous Galerkin discretization of a static anti-plane shear crack model |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2411.00021 |