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Nodally integrated implicit gradient reproducing kernel particle method for convection dominated problems. (English) Zbl 1425.65112
Summary: Convective transport terms in Eulerian conservation laws lead to numerical instability in the solution of Bubnov-Galerkin methods for these non-self-adjoint PDEs. Stabilized Petrov-Galerkin methods overcome this difficulty, however gradient terms are required to construct the test functions, which are typically expensive for meshfree methods. In this work, the implicit gradient reproducing kernel particle method is introduced which avoids explicit differentiation of test functions. Stabilization is accomplished by including gradient terms in the reproducing condition of the reproducing kernel approximation. The proposed method is computationally efficient and simplifies stabilization procedures. It is also shown that the implicit gradient resembles the diffuse derivative originally introduced in the diffuse element method in [B. Nayroles et al., Comput. Mech. 10, No. 5, 307–318 (1992; Zbl 0764.65068)], and maintains the desirable properties of the full derivative. Since careful attention must be paid to efficiency of domain integration in meshfree methods, nodal integration is examined for this class of problems, and a nodal integration method with enhanced accuracy and stability is introduced. Numerical examples are provided to show the effectiveness of the proposed method for both steady and transient problems.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35Q35 PDEs in connection with fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
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