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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, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L 2 (L 2) and L 2 (H 1) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow using the discontinuous piecewise polynomials on finite element partitions with the arbitrary shape of polygons with certain shape regularity. The fully discrete scheme is based on first order in time Euler method. We have derived O (h r + 1) in L 2 (L 2) norm and O (h r) in L 2 (H 1) norm when the exact solution u ∈ L 2 (0 , T ; H r + 1 (Ω)) ∩ H 1 (0 , T ; H r − 1 (Ω)) , for some r ≥ 1. Numerical experiments are reported for several test cases to justify our theoretical convergence results. [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]

The paper concerns numerical study of non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media with discontinuous coefficients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space–time finite element discretizations are based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L ∞ (L 2) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for non-homogeneous inner structures. [ABSTRACT FROM AUTHOR]

The numerical solution of pulsed electric interface model is discussed. A fully discrete finite element scheme based on backward Euler time discretization is proposed to approximate the voltage potential of the pulsed electric model across the physical media. Pulsed electric model arises in biological tissue when a biological cell is exposed to an electric field. Optimal pointwise‐in‐time error estimates in both L2 and H1‐norms are shown to hold for the numerical scheme. Finally, we give numerical examples to verify the theoretical results. [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]

We derive residual‐based a posteriori error estimates of finite element method for linear wave equation with discontinuous coefficients in a two‐dimensional convex polygonal domain. A posteriori error estimates for both the space‐discrete case and for implicit fully discrete scheme are discussed in L∞(L2) norm. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates in conjunction with appropriate adaption of the elliptic reconstruction technique of continuous and discrete solutions. We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the L∞(L2) norm. [ABSTRACT FROM AUTHOR]

The numerical solution of a second-order linear parabolic interface problem by weak Galerkin finite element method is discussed. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. In the implementation, the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the corresponding WG scheme is significantly impacted by the selection of such polynomials. This paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. More precisely, optimal order error estimates in both H1 and L2 norms are established for lowest order WG finite element space . Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polygonal meshes. [ABSTRACT FROM AUTHOR]

In this article a fitted finite element method is proposed and analyzed for wave equation with discontinuous coefficients. Typical semidiscrete and an implicit fully discrete schemes are presented and analyzed. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L ∞ ( L 2 ) norm. The convergence analysis relies heavily on time reconstructions of continuous and discrete solutions, in conjunction with some known results on elliptic interface problems. Finally, a numerical experiment is presented to verify our theoretical result. [ABSTRACT FROM AUTHOR]

A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L ∞( L2) and L ∞( H1) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L ∞( H1) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [ABSTRACT FROM AUTHOR]

Semidiscrete finite element approximations of the linear and semilinear parabolic interface problems are studied in the article. The convergence of the finite element solutions to the exact solutions are analyzed for fitted finite element method with straight interface triangles. Under practical regularity assumptions of the true solution it seems difficult to achieve optimal order convergence with straight interface triangles. In this exposition optimal error estimates in the L 2(L 2) and L 2(H 1) norms are established for linear semidiscrete scheme. Then an extension to the semilinear problem is also considered and related optimal error estimates are achieved. The interfaces are assumed to be smooth for our purpose. [ABSTRACT FROM AUTHOR]

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L2(Ω) and H1(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to study the finite element methods for second-order semilinear elliptic and parabolic interface problems in two dimensional convex polygonal domains. Optimal order error estimate in the -norm is proved for the semilinear elliptic interface problem when the grid lines follow the actual interface. An extension to the semilinear parabolic interface problem is considered, and both semidiscrete and fully discrete schemes are discussed. The convergence of the semidiscrete solution to the exact solution is shown to be of order in the -norm. Further, a fully discrete scheme based on backward Euler method is analyzed and optimal energy-norm error estimate is established. The interface is assumed to be of arbitrary shape but is smooth. [Copyright &y& Elsevier]

A finite-element discretization, independent of the location of the interface, is proposed and analysed for linear elliptic and parabolic interface problems. We establish error estimates of optimal order in the H1-norm and almost optimal order in the L2-norm for elliptic interface problems. An extension to parabolic interface problems is also discussed and an optimal error estimate in the L2(0, T ; H2(ω))-norm and an almost optimal order estimate in the L2(0, T ; L2(ω))-norm are derived for the spatially discrete scheme. A fully discrete scheme based on the backward Euler method is analysed and an optimal order error estimate in the L2(0, T ; H1(ω))-norm is derived. The interfaces are assumed to be of arbitrary shape and smooth for our purpose. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [ Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L 2 and H 1 norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established. [ABSTRACT FROM AUTHOR]

A finite element discretization is proposed and analyzed for a linear parabolic problems with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math., 79 (1998), pp. 175--202]. In this paper, we have used a finite element discretization, where interface triangles are assumed to be curved triangles instead of straight triangles as in classical finite element methods. Optimal order error estimates in L2 and H1 norms are shown to hold even if the regularity of the solution is low on the whole domain. While the continuous time Galerkin method is discussed for the spatially discrete scheme, the discontinuous Galerkin method is analyzed for the fully discrete scheme. The interfaces and boundaries of the domains are assumed to be smooth for our purpose. [ABSTRACT FROM AUTHOR]