Eun-Jae Park

Yonsei University · Department of Computational Science and Engineering

In this paper, C1-conforming element methods are analyzed for the stream function formulation of a single layer non-stationary quasi-geostrophic equation in the ocean circulation model. In its first part, some new regularity results are derived, which show exponential decay property when the wind shear stress is zero or exponentially decaying. More...

This paper introduces a novel staggered discontinuous Galerkin (SDG) method tailored for solving elliptic equations on polytopal meshes. Our approach utilizes a primal-dual grid framework to ensure local conservation of fluxes, significantly improving stability and accuracy. The method is hybridizable and reduces the degrees of freedom compared to...

In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Ou...

In this paper, we propose a novel staggered least squares method for elliptic equations on polygonal meshes. Our new method can be flexibly applied to rough grids and allows hanging nodes, which is of particular interest in practical applications. Moreover, it offers the advantage of not having to deal with inf-sup conditions and yielding positive...

In this paper, we propose a pressure robust staggered discontinuous Galerkin method for the Stokes equations on general polygonal meshes by using piecewise constant approximations. We modify the right-hand side of the body force in the discrete formulation by exploiting a divergence preserving velocity reconstruction operator, which is the crux for...

In this paper we propose and analyze a staggered discontinuous Galerkin method for a five-field formulation of the Biot system of poroelasticity on general polygonal meshes. Elasticity is equipped with a stress–displacement–rotation formulation with weak stress symmetry for arbitrary polynomial orders, which extends the piecewise constant approxima...

In this paper, a staggered cell-centered discontinuous Galerkin method is developed for the biharmonic problem with the Steklov boundary condition. Our approach utilizes a first-order system form of the biharmonic problem and can handle fairly general meshes possibly including hanging nodes, which favors adaptive mesh refinement. Optimal order erro...

In this paper, we present a priori and a posteriori analysis of a staggered discontinuous Galerkin (DG) method for quasi-linear second order elliptic problems of nonmonotone type. First, existence is proved by using Brouwer’s fixed point argument and uniqueness is verified utilizing Lipschitz continuity of the discrete solution map. Next, optimal a...

In this paper, we are concerned with error analysis of the semi-discrete and fully discrete approximations to the pseudostress-velocity formulation of the unsteady Stokes problem. The pseudostress-velocity formulation of the Stokes problem allows a Raviart-Thomas mixed finite element. For the semi-discrete approximation, we prove that solution oper...

In this article, we introduce and analyze arbitrary‐order, locally conservative hybrid discontinuous Galkerin methods for linearized Navier–Stokes equations. The unknowns of the global system are reduced to trace variables on the skeleton of a triangulation and the average of pressure on each cell via embedded static condensation. We prove that the...

In this paper, we present and analyze a staggered discontinuous Galerkin method for Darcy flows in fractured porous media on fairly general meshes. A staggered discontinuous Galerkin method and a standard conforming finite element method with appropriate inclusion of interface conditions are exploited for the bulk region and the fracture, respectiv...

Hybrid difference methods are a kind of finite difference methods which is similar to hybrid discontinuous Galerkin methods introduced by Jeon and Park (SIAM J. Numer. Anal., 2010). In the previous hybrid difference method, the approximate solution is only defined on the lines parallel to the coordinate axes, but the approximation is not defined at...

In this paper, nonlinear parabolic partial differential equations are considered to approximate by multiscale mortar mixed method. The key idea of the multiscale mortar mixed approach is to decompose the domain into the smaller subregions separated by the interfaces with the Dirichlet pressure boundary condition. Each subdomain is partitioned indep...

In this paper, we design and analyze staggered discontinuous Galerkin methods of arbitrary polynomial orders for the stationary Navier-Stokes equations on polygonal meshes. The exact divergence-free condition for the velocity is satisfied without any postprocessing. The resulting method is pressure-robust so that the pressure approximation does not...

In this work, we develop novel adaptive hybrid discontinuous Galerkin algorithms for second-order elliptic problems. For this, two types of reliable and efficient, modulo a data-oscillation term, and fully computable a posteriori error estimators are developed: the first one is a simple residual type error estimator, and the second is a flux recons...

In this paper, we propose and analyze a nonconforming Morley finite element method for the stationary quasi-geostrophic equation in the ocean circulation. Stability and the inf–sup condition for the discrete solution are proved, and the local existence of a unique solution to the discrete nonlinear system is established based on the assumption of t...

In this paper, we propose and analyze a nonconforming Morley finite element method for the stationary quasi-geostrophic equation in the ocean circulation. Stability and the inf-sup condition for the discrete solution are proved, and the local existence of a unique solution to the discrete nonlinear system is established based on the assumption of t...

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows hanging nodes, which...

In this paper we propose and analyze a staggered discontinuous Galerkin method for a five-field formulation of the Biot system of poroelasticity on general polygonal meshes. Elasticity is equipped with stress-displacement-rotation formulation with weak stress symmetry for arbitrary polynomial orders, which extends the piecewise constant approximati...

In this article, we develop and analyze two-grid/multi-level algorithms via mesh refinement in the abstract framework of Brezzi, Rappaz, and Raviart for approximation of branches of nonsingular solutions. Optimal fine grid accuracy of two-grid/multi-level algorithms can be achieved via the proper scaling of relevant meshes. An important aspect of t...

In this paper we propose a pressure robust staggered discontinuous Galerkin method for the Stokes equations on general polygonal meshes by using piecewise constant approximations. We modify the right hand side of the body force in the discrete formulation by exploiting divergence preserving velocity reconstruction operator, which is the crux for pr...

In this paper, we present and analyze a staggered discontinuous Galerkin method for Darcy flows in fractured porous media on fairly general meshes. A staggered discontinuous Galerkin method and a standard conforming finite element method with appropriate inclusion of interface conditions are exploited for the bulk region and the fracture, respectiv...

We consider the discretization of nonlinear second order elliptic partial differential equations by multiscale mortar expanded mixed method. This is a domain decomposition method in which the model problem is restricted to the small pieces by dividing the computational domain into the non-overlapping subdomains. An unknown (Lagrange multiplier) is...

In this paper we propose a locally conservative, lowest-order staggered discontinuous Galerkin method for the coupled Stokes–Darcy problem on general quadrilateral and polygonal meshes. This model is composed of Stokes flow in the fluid region and Darcy flow in the porous media region, coupling together through mass conservation, balance of normal...

A novel high-order hybrid staggered discontinuous Galerkin method for general meshes is proposed to solve general second order elliptic problems. Our new formulation is related to standard staggered discontinuous Galerkin method, but more flexible and cost effective: rough grids are allowed and the size of the final system is remarkably reduced tha...

This paper concerns the development of a finite-element formulation using Nitsche’ method for the phase-field model to capture an equilibrium shape of a single component vesicle. The phase-field model derived from the minimization of the curvature energy results in a nonlinear fourth-order partial differential equation. A standard conforming Galerk...

In this paper we develop a staggered discontinuous Galerkin method for the Stokes and Darcy-Forchheimer problems coupled with the \Red{Beavers-Joseph-Saffman} conditions. The method is defined by imposing staggered continuity for all the variables involved and the interface conditions are enforced by switching the roles of the variables met on the...

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general quadrilateral and polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows han...

In this work, we aim to develop efficient numerical schemes for a nonlinear fourth-order partial differential equation arising from the so-called dynamic Gao beam model. We use C⁰ interior penalty finite element methods over the spatial domain to set up the semi-discrete formulations. Convergence results for the semi-discrete case are shown, based...

In this article, we present a unified error analysis of two-grid methods for a class of nonlinear problems. We first study the two-grid method of Xu by recasting the methodology in the abstract framework of Brezzi, Rappaz, and Raviart (BRR) for approximation of branches of nonsingular solutions and derive a priori error estimates. Our convergence r...

In this paper, a locally conservative, lowest order staggered discontinuous Galerkin method is developed for the Stokes equations. The proposed method allows rough grids and is based on the partition of the domain into arbitrary shapes of quadrilaterals or polygons, which makes the method highly desirable for practical applications. A priori error...

An upwind staggered discontinuous Galerkin (upwind-SDG) method for convection dominant diffusion problems is developed. Optimal a priori error estimates can be achieved for both the scalar and vector functions approximated by the method. To efficiently capture the layer problems, we propose a robust a posteriori error estimator for upwind-SDG metho...

In this paper, we present for the first time guaranteed upper bounds for the staggered discontinuous Galerkin method for diffusion problems. Two error estimators are proposed for arbitrary polynomial degrees and provide an upper bound on the energy error of the scalar unknown and \(L^2\)-error of the flux, respectively. Both error estimators are ba...

We first propose a guaranteed upper bound for an arbitrary order staggered discontinuous Galerkin (staggered DG) method for the Stokes equations with the use of the global inf–sup constant. Equilibrated stress reconstruction and velocity reconstruction are the main ingredients in the construction of the error estimator. Next, to improve the error e...

In this paper, we are concerned with space-time a posteriori error estimators for fully discrete solutions of linear parabolic problems. The mixed formulation with Raviart–Thomas finite element spaces is considered. A new second-order method in time is proposed so that mixed finite element spaces are permitted to change at different time levels. Th...

This paper presents theoretical error estimates of B-spline based finite-element methods for the streamfunction formulation of the stationary quasi-geostrophic equations, which describe the large scale wind-driven ocean circulation. We introduce variational formulations of the streamfunction formulation inspired by the interior penalty discontinuou...

This manuscript is a corrected version of our paper that appeared in Computational Methods in Applied Mathematics 17, 253-267 (2017). Most corrections are made in Section 4, which are colored in blue.

In this paper, we first propose and analyze a locally conservative, lowest order staggered discontinuous Galerkin method of minimal dimension on general quadrilateral/polygonal meshes for elliptic problems. The method can be flexibly applied to rough grids such as the highly distorted trapezoidal grid, and both h perturbation and h² perturbation of...

In this paper, we prove optimal a priori error estimates for the pseudostress-velocity mixed finite element formulation of the incompressible Navier–Stokes equations, thus improve the result of Cai et al. (SINUM 2010). This is achieved by applying Petrov–Galerkin type Brezzi–Rappaz–Raviart theory.

A multiscale mortar mixed finite element method is established to approximate non-linear second order elliptic equations. The method is based on non-overlapping domain decomposition and mortar finite element methods. The existence and uniqueness of the approximation are demonstrated, and a priori L2-error estimates for the velocity and pressure are...

This paper presents a nonconforming finite element method for a streamfunction formulation of the stationary quasi-geostrophic equations, which describe the large scale wind-driven ocean circulation. The streamfunction formulation is a fourth order nonlinear PDE and the nonconforming method is based on C0-elements instead of C1-elements. Existence...

The first-order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional. In this paper we prove Q-linear convergence of the associated adaptive mesh-refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant...

In this work, we present novel high-order discontinuous Galerkin methods with Lagrange multiplier (DGLM) for hyperbolic systems of conservation laws. Lagrange multipliers are introduced on the inter-element boundaries via the concept of weak divergence. Static condensation on element unknowns considerably reduces globally coupled degrees of freedom...

A locally conservative, hybrid spectral difference method (HSD) is presented and analyzed for the Poisson equation. The HSD is composed of two types of finite difference approximations; the cell finite difference and the interface finite difference. Embedded static condensation on cell interior unknowns considerably reduces the global couplings, re...

This book is a collection of papers presented at the 23rd International Conference on Domain Decomposition Methods in Science and Engineering, held on Jeju Island, Korea on July 6-10, 2015. Domain decomposition methods solve boundary value problems by splitting them into smaller boundary value problems on subdomains and iterating to coordinate the...

A locally conservative, hybrid spectral difference method (HSD) is presented and analyzed for the Poisson equation. The HSD is composed of two types of finite difference approximations; the cell finite difference and the interface finite difference. Embedded static condensation on cell interior unknowns considerably reduces the global couplings, re...

A locally conservative, hybrid spectral difference method (HSD) is presented and analyzed for the Poisson equation. The HSD is composed of two types of finite difference approximations; the cell finite difference and the interface finite difference. Embedded static condensation on cell interior unknowns considerably reduces the global couplings, re...

This work concerns the development of a finite-element algorithm for the stationary quasi-geostrophic equations to treat the large scale wind-driven ocean circulation. The algorithm is developed based on the streamfunction formulation involving fourth-order gradients of the streamfunction. Here, we examine the adaptation of a relatively inexpensive...

Optimal control problems, governed by convection–diffusion equations with bilinear control, are studied. For the realization of the numerical solution, the multigrid for optimization method together with finite difference discretization is utilized and investigated. In addition, the extension to constrained optimal control problems with bilinear co...

A new asymptotically exact a posteriori error estimator is developed for first-order div least-squares (LS) finite element methods. Let be LS approximate solution for . Then, is asymptotically exact a posteriori error estimator for or depending on the order of approximate spaces for and . For to be asymptotically exact for , we require higher order...

The first-order div least squares finite element methods (LSFEMs) allow for an immediate a posteriori error control by the computable residual of the least squares functional. This paper establishes an adaptive refinement strategy based on some equivalent refinement indicators. Since the first-order div LSFEM measures the flux errors in H (div), th...

A hybrid discontinuous Galerkin (HDG) method for the Poisson problem introduced by Jeon and Park can be viewed as a hybridizable discontinuous Galerkin method using a Baumann-Oden type local solver. In this work, an upwind HDG method with super-penalty is proposed to solve advection-diffusion-reaction problems. A super-penalty formulation facilitat...

In this work, we consider mathematical and numerical approaches to a dynamic contact problem with a highly nonlinear beam, the so-called Gao beam. Its left end is rigidly attached to a supporting device, whereas the other end is constrained to move between two perfectly rigid stops. Thus, the Signorini contact conditions are imposed to its right en...

In this paper, we propose an effective iterative preconditioning method to solve elliptic problems with jumps in coefficients. The algorithm is based on the additive Schwarz method (ASM). First, we consider a domain decomposition method without 'cross points' on interfaces between subdomains and the second is the 'cross points' case. In both cases...

A locally conservative hybridized finite element method for Stokes equations is presented and analyzed. The hybridized approach reduces a lot of degrees of freedom, especially for pressure approximation. In our approach the pressure is determined locally up to a constant, therefore, the global stiffness system contains only the average of pressure...

In this article, we propose and analyze a new nonconforming primal mixed finite element method for the stationary Stokes equations. The approximation is based on the pseudostress-velocity formulation. The, incompressibility condition is used to eliminate the pressure variable in terms of trace-free pseudostress. The pressure is then computed from a...

We propose and analyze two-scale product approximation for semilinear heat equations in the mixed finite element method. In order to efficiently resolve nonlinear algebraic equations resulting from the mixed method for semilinear parabolic problems, we treat the nonlinear terms using some interpolation operator and exploit a two-scale grid algorith...

An upstream scheme based on the pseudostress-velocity mixed formulation is studied to solve convection-dominated Oseen equations. Lagrange multipliers are introduced to treat the trace-free constraint and the lowest order Raviart-Thomas finite element space on rectangular mesh is used. Error analysis for several quantities of interest is given. Par...

In the field of uncertainty quantification (UQ), epistemic uncertainty often refers to the kind of uncertainty whose complete probabilistic description is not available, largely due to our lack of knowledge about the uncertainty. Quantification of the impacts of epistemic uncertainty is naturally difficult, because most of the existing stochastic t...

A new family of locally conservative, finite element methods for a rectangular mesh is introduced to solve second-order elliptic equations. Our approach is composed of generating PDE-adapted local basis and solving a global matrix system arising from a flux continuity equation. Quadratic and cubic elements are analyzed and optimal order error estim...

The stress-velocity formulation of the stationary Stokes problem allows an Arnold-Winther mixed finite element formulation with some superconvergent reconstruction of the velocity. This local postprocessing gives rise to two reliable a posteriori error estimators which recover optimal convergence order for the stress error estimates. The theoretica...

The pseudostress-velocity formulation of the stationary Stokes problem allows a Raviart-Thomas mixed finite element formulation with quasi-optimal convergence and some superconvergent reconstruction of the velocity. This local postprocessing gives rise to some averaging a posteriori error estimator with explicit constants for reliable error control...

The pseudostress-velocity formulation of the stationary Stokes problem allows a Raviart-Thomas mixed finite element formulation with quasi-optimal convergence and some superconvergent reconstruction of the velocity. This local postprocessing gives rise to some averaging a posteriori error estimator with explicit constants for reliable error control...

In this paper, we propose a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method. In particular, we propose an a posteriori error estimator for when one is interested in ‖σ−σh‖0 where σ=−A∇u. Our a posteriori error estimators are obtained by assigning proper weight (in terms of local m...

We study the mixed flnite element approximation of the second-order elliptic problem with gradient nonlinearities. Existence and uniqueness of the approximate solution are proved and optimal order a priori error estimates in Lm(›) are obtained. Also, reliable and e-cient a posteriori error estimators measured in the Lm(›)-norm are derived. 1. Intro...

A new family of hybrid discontinuous Galerkin methods is studied for second-order elliptic equations. Our proposed method is a generalization of the cell boundary element (CBE) method [Y. Jeon and E.-J. Park, Appl. Numer. Math., 58 (2008), pp. 800-814], which allows high order polynomial approximations. Our method can be viewed as a hybridizable di...

A numerical method is proposed and analyzed to approximate a mathematical model of age-dependent population dynamics with spatial diffusion. The model takes a form of nonlinear and nonlocal system of integro-differential equations. A finite difference method along the characteristic age-time direction is considered and primal mixed finite elements...

We consider the model second-order elliptic problem: r Kru = f in ; u = 0 on @ ; where is a bounded polygonal domain in R2. Assume that is composed of disjoint polygonal subdomains 1; ; J and that K is a function such that 0 < K K(x) K < 1 and K(x) = Kj in j for each j. The localized problem becomes KT u = f in T;

We construct a posteriori error estimators for approximate solutions of linear parabolic equations. We consider discretizations of the problem by modified discontinuous Galerkin schemes in time and continuous Galerkin methods in space. Especially, finite element spaces are permitted to change at different time levels. Exploiting Crank-Nicolson reco...

The nonconforming cell boundary element (CBE) methods are proposed. The methods are designed in such a way that they enjoy the mass conservation at the element level and the normal component of fluxes at inter-element boundaries are continuous for unstructured triangular meshes. Normal flux continuity and the optimal order error estimates in a brok...

In this paper, we design a posteriori error estimators for the mixed finite element approximation of the convection–diffusion problems with dominant convection which exhibit solutions with internal layers and boundary layers. A discontinuous upstream weighting scheme in association with the mixed method is used for the convection term. The a poster...

We summarize the results on the cell boundary element methods (CBE methods) and the multiscale cell boundary element method based on papers by Y. Jeon and D. Sheen [Adv. Comput. Math. 22, No. 3, 201–222 (2005; Zbl 1067.65138)], Y. Jeon, E.-J. Park, and D. Sheen [Numer. Methods Partial Differ. Equations 21, No. 3, 496–511 (2005; Zbl 1072.65158)], an...

We consider multiscale mortar mixed finite element discreti-zations for slightly compressible Darcy flows in porous media. This paper is an extension of the formulation introduced by Arbogast et al. for the incompressible problem [2]. In this method, flux continuity is imposed via a mortar finite element space on a coarse grid scale, while the equa...

In this article, we construct an a posteriori error estimator for expanded mixed hybrid finite-element methods for second-order elliptic problems. An a posteriori error analysis yields reliable and efficient estimate based on residuals. Several numerical examples are presented to show the effectivity of our error indicators. © 2006 Wiley Periodical...

In this paper, we consider second order linear elliptic and parabolic equations that model single phase Darcy flows in porous media. First, we study multiscale mortar mixed finite element discretizations for model elliptic prob-lems introduced by Arbogast et al.. This approach is based on domain decomposition theory and mortar finite elements. In t...

The purpose of the paper is to introduce a novel cell boundary element (CBE) method for the convection dominated diffusion equation. The CBE method can be viewed as a Petrov-Galerkin type method defined on the skeleton of a mesh. The proposed method utilizes continuity of normal flux on each inter-element boundary. By constructing a local basis (me...

We study the primal mixed finite-element approximation of the second-order elliptic problem with gradient nonlinearities. Existence and uniqueness of the approximate solution are proved and optimal-order a priori error estimates are obtained. Also, reliable a posteriori error estimators are derived.

An elementary analysis on the cell boundary element (CBEM) was given by Jeon and Sheen. In this article we improve the previous results in various aspects. First of all, stability and convergence analysis on the rectangular grids are established. Moreover, error estimates are improved. Our improved analysis was possible by recasting of the CBEM in...

Magnetic resonance electrical impedance tomography (MREIT) is a new conductivity imaging modality that was motivated to deal with the well-known ill-posedness problem in electrical impedance tomography (EIT). In order to bypass this ill-posed nature, MREIT takes advantage of an MRI scanner as a tool to capture the z-component Bz of the induced inte...

Mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the flow of a single-phase fluid in a porous medium in ℝd, d ≤ 3, subject to Forchhheimer's law—a nonlinear form of Darcy's law. Existence and uniqueness of the approximation are proved, and optimal order error estimates in L∞(J...

The objective of numerical analysis is to devise and an-alyze efficient algorithms or numerical methods for equations arising in mathematical modeling for science and engineering. In this ar-ticle, we present some recent topics in computational mathematics, specially in the finite element method and overview the development of the mixed finite elem...

In this paper we extend the mixed flnite element method and its L2¡error estimate for postprocessed solutions by using Crank- Nicolson time-discretization method. Global O(h2+(¢t)2)-superconvergence for the lowest order Raviart- Thomas element (Q0 ¡Q1;0 £Q0;1) are obtained. Numerical exam- ples are presented to conflrm superconvergence phenomena.

The main focus of this paper is to suggest a domain decomposition method for mixed finite element approximations of elliptic problems with anisotropic coefficients in domains. The theorems on traces of functions from Sobolev spaces play an important role in studying boundary value problems of partial differential equations. These theorems are commo...

A new image reconstruction algorithm is proposed to visualize static conductivity images of a subject in magnetic resonance electrical impedance tomography (MREIT). Injecting electrical current into the subject through surface electrodes, we can measure the induced internal magnetic flux density B = (Bx, By, Bz) using an MRI scanner. In this paper,...

This paper is concerned with the analysis of a generalized Gurtin-MacCamy model describing the evolution of an age-structured population. The problem of global boundedness is studied. Namely we ask whether there are simple general assumptions that one can make on the vital rates in order to have boundedness of the solution. Next a fully implicit fi...

Fully discrete mixed finite element method is considered to approximate the solution of a nonlinear second-order parabolic problem. A massively parallel iterative procedure based on domain decomposition technique is presented to solve resulting nonlinear algebraic equations. Robin type boundary conditions are used to transmit information between su...

We propose a numerical procedure to solve the equations describing non-Darcy flow of a single-phase fluid in a porous medium in two or three spacial dimensions, including the generalized Forchheimer equation. Fully discrete mixed finite element methods are considered and analyzed for the approximation. Existence and uniqueness of the approximation...

A numerical approximation is considered for a model of epidemiology describing age-dependent population dynamics with spatial diffusion. A finite difference method along the characteristic age-time direction is combined with finite elements in the spatial variable. Optimal order error estimates are derived for the approximation.

Here we investigate the optimal harvesting problem for some periodic age-dependent population dynamics; namely, we consider the linear Lotka--McKendrick model with periodic vital rates and a periodic forcing term that sustains oscillations. Existence and uniqueness of a positive periodic solution are demonstrated and the existence and uniqueness of...

Splitting methods for nonlinear models arising from population dynamics and epidemiology are described and analyzed. A backward finite differencing along the characteristic is used for the approximation. It is shown that the schemes are convergent at first-order rate in the maximum norm. The stability of the methods is discussed. Several numerical...

The numerical solution of Dirichlet's problem for a second-order elliptic operator in divergence form with arbitrary nonlinearities in the first-and zero-order terms is considered. The mixed finite-element method is used. Existence and uniqueness of the approximation are proved and optimal error estimates in L2 are demonstrated for the relevant fun...

Mixed finite element methods are considered to approximate the solution of fully nonlinear second order parabolic problems in divergence form in Rd, d ≤ 3. Existence and uniqueness of the approximation are proved. Optimal order error estimates in L∞ (J; L2(Ω)) and in are demonstrated for the relevant variables.

The p-version of the mixed finite element method is considered for nonlinear second-order elliptic problems. Existence and uniqueness of the approximation are demonstrated and optimal order error estimates in L2 are derived for the three relevant functions. Error estimates for the scalar function are also given in Lq, 2 ⩽ q ⩽ + ∞. © 1996 John Wiley...

Mixed finite element methods for treating the Dirichlet problem for fully nonlinear second-order elliptic operators in divergence form are extended to cover the three-dimensional case. Existence and uniqueness of the approximation are proved, and optimal error estimates in L2 are demonstrated for both the scalar and vector functions approximated by...

Fully discrete mixed finite element methods are presented for the approximation of the solution of the system of equations that describes the flow of a single-phase fluid in a porous medium in IRd, d ≤ 3, subject to Forchheimer's law - a nonlinear form of Darcy's law. Optimal order error estimates in L2 are given for the three relevant functions. N...

A numerical method is proposed to approximate the solution of a nonlinear and nonlocal system of integro-differential equations describing age-dependent population dynamics with spatial diffusion. A finite difference method along the characteristic age-time direction combined with mixed finite elements in the spatial variable is used for the approx...