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In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝd (d = 2, 3). This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ≥ 1 for the velocity and polynomials of degree k – 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips. [ABSTRACT FROM AUTHOR]

In this article, we propose the weak Galerkin (WG) finite element schemes for H(div;Ω)${\bf H}(\mbox{div}; {\Omega })$‐elliptic problems with and without stabilizers. Optimal orders of convergence are established for the WG approximations in both discrete energy norm and L2 norm. Removing stabilizers from WG finite element methods will simplify the formulations, reduce programming complexity, and may also speed up the computation time. More precisely, for sufficiently smooth solutions, we have proved the supercloseness of order two for the stabilizer free weak Galerkin finite element solution. Several numerical tests are presented to demonstrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In the present work, we have described a systematic numerical study on weak Galerkin (WG) finite element method for second‐order linear parabolic problems by allowing polynomial approximations with various degrees for each local element. Convergence of both semidiscrete and fully discrete WG solutions are established in L∞L2$$ {L}^{\infty}\left({L}^2\right) $$ and L∞H1$$ {L}^{\infty}\left({H}^1\right) $$ norms for a general WG element 풫k(K),풫j(∂K),풫l(K)2, where k≥1$$ k\ge 1 $$, j≥0$$ j\ge 0 $$ and l≥0$$ l\ge 0 $$ are arbitrary integers. The fully discrete space–time discretization is based on a first order in time Euler scheme. Numerical experiments are reported to justify the robustness, reliability and accuracy of the WG finite element method. [ABSTRACT FROM AUTHOR]

In this study, we consider a stabilizer free weak Galerkin (SFWG) finite element method to solve a second‐order Sobolev equation. The SFWG method has various assets, including the support for higher order of accuracy and fewer coefficients. We have analyzed error estimates for both semidiscrete and fully discrete schemes using a non‐standard projection operator. The convergence rate of numerical error estimates of Ohk+2+τ2$$ \kern0.3em O\kern0.3em \left({h}^{k+2}+{\tau}^2\right) $$ in L∞H1$$ {L}^{\infty}\left({H}^1\right) $$ norm and Ohk+3+τ2$$ \kern0.3em O\kern0.3em \left({h}^{k+3}+{\tau}^2\right) $$ in L∞L2$$ {L}^{\infty}\left({L}^2\right) $$ norm are obtained, which are two order higher than the optimal order associated with WG finite element space ℙk(K),ℙk+1(∂K),ℙk+1(K)2.$$ \left({\mathrm{\mathbb{P}}}_k(K),{\mathrm{\mathbb{P}}}_{k+1}\left(\partial K\right),{\left[{\mathrm{\mathbb{P}}}_{k+1}(K)\right]}^2\right). $$ At last, several numerical examples are reported to validate the supercloseness property and efficiency of the proposed SFWG method. These experiments confirm the accuracy of the theoretical findings. [ABSTRACT FROM AUTHOR]

We analyze the weak Galerkin finite element methods for second-order linear parabolic problems with L² initial data, both in a spatially semidiscrete case and in a fully discrete case based on the backward Euler method. We have established optimal L² error estimates of order O(h²=t) for semisdiscrete scheme. Subsequently, the results are extended for fully discrete scheme. The error analysis has been carried out on polygonal meshes for discontinuous piecewise polynomials in finite element partitions. Finally, numerical experiments confirm our theoretical convergence results and efficiency of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we analyze finite element methods applied to pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. Considering the cell to be a conductive body, embedded in a more or less conductive medium, the governing system involves an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. A fitted finite element method with straight interface triangles is proposed to approximate the voltage of the pulsed electric model across the physical media. Optimal pointwise-in-time error estimates in L 2 -norm and H 1 -norm are shown to hold for semidiscrete scheme even if the regularity of the solution is low on the whole domain. Further, a fully discrete approximation based on Crank–Nicolson scheme is analyzed and related optimal error estimates are derived. [ABSTRACT FROM AUTHOR]

In this paper, the weak Galerkin finite element method (WG‐FEM) is applied to a pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. A fitted WG‐FEM is proposed to approximate the voltage of the pulsed electric model across the physical media involving an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Optimal pointwise‐in‐time error estimates in L2‐norm and H1‐norm are shown to hold for the semidiscrete scheme even if the regularity of the solution is low on the whole domain. Furthermore, a fully discrete approximation based on backward Euler scheme is analyzed and related optimal error estimates are derived. [ABSTRACT FROM AUTHOR]

We propose a fitted finite element method for non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media that have a heterogeneous conductivity. Well-posedness of the model interface problem is established. A continuous piecewise linear finite element space is employed for the spatially semidiscrete approximation and the temporal discretization is based on backward scheme. Optimal order error estimates for both semidiscrete and fully discrete schemes are proved in L ∞ (H 1) norm. Finally, we give numerical examples to verify our theoretical results. The new results and finite element schemes can be applied in the fields of engineering, medicine, and biotechnology. [ABSTRACT FROM AUTHOR]