Uniform superconvergent analysis of a new mixed finite element method for nonlinear Bi-wave singular perturbation problem

Quasi‐uniform convergence analysis of a modified penalty finite element method for nonlinear singularly perturbed bi‐wave problem

“…Proof. We start to prove the existence of the solution h of problem (5). In fact, from (3) and (5), we get…”

“…We prove the well‐posedness of the approximation solution through Brouwer's fixed point theorem and derive optimal error estimate in the energy norm independent of the negative powers of the real perturbation parameter δ , that is, quasi‐uniform behavior. In particular, the nonlinear term of problem (1) is dealt with rigorously through a splitting technique [5] based on the monotonically increasing character of f ( ψ ) in the whole error analysis. It seems that optimal error estimate of the proposed method has improved the corresponding conclusion of [3].…”

“…For example, two conforming Galerkin finite element methods (FEMs) and the modified nonconforming Morley-type Galerkin FEMs were proposed for the linear bi-wave equations, and convergence error estimates were derived in [2,3], respectively. The Ciarlet-Raviart mixed FEMs were analyzed in [4][5][6] for the linear bi-wave equations and problem (1), and the uniform superconvergence estimates in the broken H 1 norm were deduced, respectively. We mention that the piecewise polynomials space was required to be C 1 continuous in [2], which would be expensive and less efficient.…”

Uniform Superconvergence Analysis of a Two-Grid Mixed Finite Element Method for the Time-Dependent Bi-Wave Problem Modeling $D$-Wave Superconductors

“…Hence, 0 < δ < 1 is expected to be small for d-wave superconductors and problem (1.1) degenerates into the semilinear parabolic equation when δ → 0. In recent years, there are some theoretical analysis and numerical simulations about FEMs, such as optimal order error estimates of conforming Galerkin FEMs and the modified Morleytype discontinuous Galerkin FEMs in [10,11], uniform superconvergence error estimates of Ciarlet-Raviart schemes with the conforming and nonconforming elements in [12][13][14]. But these work mainly focused on the stationary singularly perturbed Bi-wave problems.…”

New High Accuracy Analysis of a Double Set Parameter Nonconforming Element for the Clamped Kirchhoff Plate Unilaterally Constrained by an Elastic Obstacle

“…Compared with the conforming FEMs studied in [1,11], C 0 continuous and discontinuous nonconforming (i.e., C 0 and non-C 0 nonconforming) FEMs were discussed in [10], and convergence analyses were established. On the other hand, the high accuracy analysis of nonconforming FEMs has been an active research area in practical computations, and many high accuracy results have been derived for variational inequalities and the boundary value problem [12][13][14][15][16][17][18][19][20]. However, for the plate obstacle problem (4) with a rigid obstacle, the exact solution u only belongs to H 3 (Ω) ∩ C 2 (Ω) instead of H 4 loc (Ω) [21,22]; thus, the lack of H 4 regularity makes it impossible to develop high accuracy analysis.…”