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In this paper, we consider a two-layer Crank–Nicolson scheme combined with linear finite element approximation for second-order hyperbolic optimal control problems with control constraints. For state and co-state variables, their time derivatives are first reduced to a first-order system then discretized by the two-layer Crank–Nicolson scheme and their space derivatives are discretized by linear finite elements. The control variable is dealt with variational discretization technique. Optimal priori error estimates for state, co-state and control variables are derived. Some numerical examples are presented to illustrate the theoretical convergence rate.

This paper studies the benefts of pressure-robust discretizations in the scope of optimal control of incompressible fows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their L²-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions lead to qualitatively better a priori estimates in the sense that the discrete velocities do not depend on the pressures scaled by the inverse of the viscosity. The consequences of the space discretization are also demonstrated and validated in numerical examples. [ABSTRACT FROM AUTHOR]

In this work, we numerically study a problem including several dissipative mechanisms. A particular case involving the symmetry of the coupling matrix and three mechanisms is considered, leading to the exponential decay of the corresponding solutions. Then, a fully discrete approximation of the general case in two dimensions is introduced by using the finite element method and the implicit Euler scheme. A priori error estimates are obtained and the linear convergence is derived under some appropriate regularity conditions on the continuous solution. Finally, some numerical simulations are performed to illustrate the numerical convergence and the behavior of the discrete energy depending on the number of dissipative mechanisms. [ABSTRACT FROM AUTHOR]

In this work, a priori error estimates for finite element approximations to the governing equations of heat and mass transfer in fluidized beds are derived. These equations are time dependent strongly coupled system of five semilinear convection-diffusion-reaction equations. The a priori error estimates for all the five variables are obtained for the error measured in L∞(L2) and L 2 (E) , E is the energy norm. [ABSTRACT FROM AUTHOR]

In this work, we consider a multi-dimensional dual-phase-lag problem arising in porous-thermoelasticity with microtemperatures. An existence and uniqueness result is proved by applying the semigroup of linear operators theory. Then, by using the finite element method and the Euler scheme, a fully discrete approximation is numerically studied, proving a discrete stability property and a priori error estimates. Finally, we perform some numerical simulations to demonstrate the accuracy of the approximation and the behavior of the solution in one- and two-dimensional problems.

In this paper, a fully discrete interpolated coefficient characteristic finite element approximation is proposed for optimal control problems governed by time-dependent semilinear convection–diffusion equations, where the hyperbolic part of the state equation is first treated by directional derivatives and then discretized by backward difference, the semilinear term is dealt with interpolation coefficient finite elements technique. A priori error estimates for the control, state and co-state variables are derived. Theoretic results are confirmed by a numerical example.

This article considers a coupled finite and boundary element method for an interface problem for the acoustic wave equation. Well-posedness, a priori and a posteriori error estimates are discussed for a symmetric space-time Galerkin discretization related to the energy. Numerical experiments in three dimensions illustrate the performance of the method in model problems. [ABSTRACT FROM AUTHOR]

In this paper, continuous space-time finite element methods are developed for approximating a class of quasilinear parabolic problems in space and in time, simultaneously. The basis of the whole approach is on a space-time variational formulation where streamline upwind terms and interface jump terms are further added for stabilizing the discretization in the direction of time. Error estimates are shown and are verified numerically through a series of numerical tests. Emphasis is placed on investigating the asymptotic convergence of the error parts, which are related to the time discretization. [ABSTRACT FROM AUTHOR]

We consider a one-dimensional two-temperatures thermoelastic model. The corresponding variational problem leads to a coupled system which is written in terms of the mechanical velocity, the temperature speed and the inductive temperature. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A priori error estimates are proved and the linear convergence of the approximations is deduced under suitable additional regularity conditions. Finally, some numerical simulations are shown to demonstrate the accuracy of the proposed algorithm and the behavior of the discrete energy. [ABSTRACT FROM AUTHOR]

In this paper, a thermomechanical problem involving a viscoelastic Timoshenko beam is analyzed from a numerical point of view. The so-called thermodiffusion effects are also included in the model. The problem is written as a linear system composed of two second-order-in-time partial differential equations for the transverse displacement and the rotational movement, and two first-order-in-time partial differential equations for the temperature and the chemical potential. The corresponding variational formulation leads to a coupled system of first-order linear variational equations written in terms of the transverse velocity, the rotation speed, the temperature and the chemical potential. The existence and uniqueness of solutions, as well as the energy decay property, are stated. Then, we focus on the numerical approximation of this weak problem by using the implicit Euler scheme to discretize the time derivatives and the classical finite element method to approximate the spatial variable. A discrete stability property and some a priori error estimates are shown, from which we can conclude the linear convergence of the approximations under suitable additional regularity conditions. Finally, some numerical simulations are performed to demonstrate the accuracy of the scheme, the behavior of the discrete energy decay and the dependence of the solution with respect to some parameters.

In this work, we consider a multi-dimensional dual-phase-lag problem arising in porous-thermoelasticity with microtemperatures. An existence and uniqueness result is proved by applying the semigroup of linear operators theory. Then, by using the finite element method and the Euler scheme, a fully discrete approximation is numerically studied, proving a discrete stability property and a priori error estimates. Finally, we perform some numerical simulations to demonstrate the accuracy of the approximation and the behavior of the solution in one- and two-dimensional problems. [ABSTRACT FROM AUTHOR]

This paper discusses some a priori error estimates of bilinear elliptic optimal control problems based on the finite volume element approximation. A case-based numerical example serves to discuss with optimal $ L^2 $-norm error estimates and $ L^{\infty} $-norm error estimates, and supports two key insights. First, the approximate orders for the state, costate and control variables are $ O(h^2) $ in the sense of $ L^{2} $-norm. Second, the approximate orders for the state, costate and control variables are $ O(h^2\sqrt{|lnh|}) $ in the sense of $ L^{\infty} $-norm.

In this paper, a strain gradient thermoelastic problem is studied from the numerical point of view. The heat conduction is modeled by using the type II thermal law and the second gradient of the thermal displacement is also included in the set of independent constitutive variables. An existence and uniqueness result is recalled. Then, the fully discrete approximations are introduced by using the implicit Euler scheme and the finite element method. A discrete stability property and a main a priori error estimates results are proved. Then, some numerical simulations are performed, including some issues as the numerical convergence of the approximations, the effect of two possible dissipative terms (second- and fourth-order) or a comparison with the type II strain gradient thermoelasticity. [ABSTRACT FROM AUTHOR]

In this paper, we analyze, from the numerical point of view, a swelling porous thermo-elastic problem. The so-called second-sound effect is introduced and modeled by using the simplest Maxwell–Cattaneo law. This problem leads to a coupled system which is written by using the displacements of the fluid and the solid, the temperature and the heat flux. The numerical analysis of this problem is performed applying the classical finite element method with linear elements for the spatial approximation and the backward Euler scheme for the discretization of the time derivatives. Then, we prove the stability of the discrete solutions and we provide an a priori error analysis. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximations, the exponential decay of the discrete energy and the dependence on a coupling parameter.

In this paper we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the nodal values inside the domain. We show that the mass matrix associated with the extended finite element space can be lumped leading to a fully explicit scheme. We derive stability estimates for the method and provide optimal order a priori error estimates. Finally, we present some illustrating numerical examples. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal L 2 $L^{2}$ -norm error estimates. The approximate orders for the state, costate, and control variables are O ( h 2 ) $O(h^{2})$ in the sense of L 2 $L^{2}$ -norm. Furthermore, we derive H 1 $H^{1}$ -norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

Optimal control problems governed by partial differential equations have become a very active and successful research area. So, in this paper, we analyzed a priori and a posteriori error estimates for the hpfinite element discretization of elliptic Robin boundary control problems. With the discrete and continuous optimality conditions of the problem, we constructed the error estimators. Based on the residual of the model equations for the coupled state and control approximations, the upper error bound is proved using Scott–Zhang-type quasi interpolation estimates. In order to provide the optimality, lower error bound is shown using some polynomial inverse estimates in weighted Sobolev spaces. Such estimators can be used to construct reliable adaptive methods for optimal control problems.

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the C0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side F ∈ H−2(Ω) replaced by F ○ (JIM) and then are quasi-optimal in their respective discrete norms. The smoother JIM is defined for a piecewise smooth input function by a (generalized) Morley interpolation IM followed by a companion operator J. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj, ESAIM: M2AN49 (2015) 977–990.] without data oscillations. This paper extends the work [Veeser and Zanotti, SIAM J. Numer. Anal.56 (2018) 1621–1642] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms. [ABSTRACT FROM AUTHOR]

We adopt the integral definition of the fractional Laplace operator and analyze an optimal control problem for a fractional semilinear elliptic partial differential equation (PDE); control constraints are also considered. We establish the well-posedness of fractional semilinear elliptic PDEs and analyze regularity properties and suitable finite element discretizations. Within the setting of our optimal control problem, we derive the existence of optimal solutions as well as first and second order optimality conditions; regularity estimates for the optimal variables are also analyzed. We devise a fully discrete scheme that approximates the control variable with piecewise constant functions; the state and adjoint equations are discretized with continuous piecewise linear finite elements. We analyze convergence properties of discretizations and derive a priori error estimates. [ABSTRACT FROM AUTHOR]

We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 {L_{2}} -norm regularization is replaced by the H - 1 {H^{-1}} -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ ⁢ h {u_{\varrho h}} to the state u and the desired state u ¯ {\overline{u}} in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 {\varrho=h^{2}} ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ {\overline{u}}. The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver. [ABSTRACT FROM AUTHOR]

This paper is concerned with conforming finite element discretizations for quasilinear elliptic problems in divergence form, of a class that generalizes the p -Laplace equation and allows to show existence and uniqueness of the continuous and discrete problems. We derive discretization error estimates under general regularity assumptions for the solution and using high order polynomial spaces, resulting in convergence rates that are then verified numerically. A key idea of this error analysis is to consider carefully the relation between the natural W 1 , p -seminorm and a specific quasinorm introduced in the literature. In particular, we are able to derive interpolation estimates in this quasinorm from known interpolation estimates in the W 1 , p -seminorm. We also give a simplified proof of known near-best approximation results in W 1 , p -seminorm starting from the corresponding result in the mentioned quasinorm. [ABSTRACT FROM AUTHOR]

We investigate the isogeometric analysis approach based on the extended Catmull–Clark subdivision for solving the PDEs on surfaces. As a compatible technique of NURBS, subdivision surfaces are capable of the refinability of B-spline techniques, and overcome the major difficulties of the interior parameterization encountered by the isogeometric analysis. The surface is accurately represented as the limit form of the extended Catmull–Clark subdivision, and remains unchanged throughout the h-refinement process. The solving of the PDEs on surfaces is processed on the space spanned by the Catmull–Clark subdivision basis functions. In this work, we establish the interpolation error estimates for the limit form of the extended Catmull–Clark subdivision function space on surfaces. We apply the results to develop the approximation estimates for solving multiple second-order PDEs on surfaces, which are the Laplace–Beltrami equation, the Laplace–Beltrami eigenvalue equation and the time-dependent Cahn–Allen equation. Numerical experiments confirm the theoretical results and are compared with the classical linear finite element method to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we consider the convection-diffusion-reaction problem coupled the Darcy-Forchheimer problem by a non-linear external force depending on the concentration. We establish existence of a solution by using a Galerkin method and we prove uniqueness. We introduce and analyse a numerical scheme based on the finite element method. An optimal a priori error estimate is then derived for each numerical scheme. Numerical investigation are performed to confirm the theoretical accuracy of the discretization. [ABSTRACT FROM AUTHOR]

In this paper, we consider mixed finite element approximations of a class of nonlinear parabolic equations. The backward Euler scheme for temporal discretization is used. Firstly, the new mixed projection is defined and the related a priori error estimates are proved. Secondly, the optimal a priori error estimates for the pressure variable and the velocity variable are derived. Finally, a numerical example is presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents the numerical analysis for a variant of the nonlinear multiscale Dynamic Diffusion (DD) method for the advection-diffusion-reaction equation initially proposed by Arruda et al. [1] and recently studied by Valli et al. [2]. The new DD method, based on a two-scale approach, introduces locally and dynamically an extra stability through a nonlinear operator acting in all scales of the discretization. We prove existence of discrete solutions, stability, and a priori error estimates. We theoretically show that the new DD method has convergence rate of O (h 1 / 2) in the energy norm, and numerical experiments have led to optimal convergence rates in the L 2 (Ω) , H 1 (Ω) , and energy norms. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the a priori error analysis of finite element method for parabolic optimal control problem with measure data in a bounded convex domain. The solution of the state equation of this kind of problem exhibits low regularity which introduces some difficulties for both theory and numerics of finite element method. We first prove the existence, uniqueness and regularity results for the solutions of control problem under low regularity assumption on the state variable. For numerical approximations, we use continuous piecewise linear functions for the state and co-state variables, and piecewise constant functions for the control variable. Both spatially discrete and fully discrete finite element approximations of the control problem are analyzed. We derive a priori error estimates of order O (h) for the state, co-state and control variables for the spatially discrete problem. A time discretization scheme based on implicit backward-Euler method is applied to obtain error estimates of order O (h + k 1 / 2) for the state, co-state and control variables. Numerical results are presented to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, we study a thermoelastic Bresse system from both mathematical and numerical points of view. The dual-phase-lag heat conduction theory is used to model the heat transfer. An existence and uniqueness result is obtained by using the theory of linear semigroups. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A priori error estimates are shown, from which the linear convergence is derived under suitable regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution with respect to a constitutive parameter. [ABSTRACT FROM AUTHOR]

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the weak singularity, fractional order integration in time is managed approximately by linear interpolation so that we can formulate a fully discrete problem. In this paper, we present a stability bound as well as a priori error estimates. Furthermore, we carry out numerical experiments with varying regularity of exact solutions at the end. [ABSTRACT FROM AUTHOR]

We analyze, on two dimensional polygonal domains, classical low–order inf-sup stable finite element approximations of the stationary Navier–Stokes equations with singular sources. We operate under the assumptions that the continuous and discrete solutions are sufficiently small. We perform an a priori error analysis on convex domains. On Lipschitz, but not necessarily convex, polygonal domains, we design an a posteriori error estimator and prove its global reliability. We also explore efficiency estimates. We illustrate the theory with numerical tests. [ABSTRACT FROM AUTHOR]

This work is motivated by recent interest in the topic of pointwise tracking type optimal control problems for the Stokes problem. Pointwise tracking consists of point evaluations in the objective functional which lead to Dirac measures appearing as source terms of the adjoint problem. Considering bounds for the control allows for improved regularity results for the exact solution and improved approximation error estimates of its numerical counterpart. We show a sub-optimal convergence result in three dimensions that nonetheless improves the results known from the literature. Finally, we offer supporting numerical experiments and insights towards optimal approximation error estimates. [ABSTRACT FROM AUTHOR]

In this paper, we study, from both an analytical and a numerical point of view, a poro-thermoelastic problem with microtemperatures. The so-called Moore–Gibson–Thompson equation is used to model the contribution for the temperature and microtemperatures. An existence and uniqueness result is proved by using the theory of linear semigroups of contractions and, for the one-dimensional case, the exponential energy decay is found under some conditions on the constitutive coefficients. Then, a fully discrete approximation is introduced by using the finite element method and the implicit Euler scheme. We show that the discrete energy decays and we obtain some a priori error estimates from which, under some adequate additional regularity conditions on the continuous solution, we derive the linear convergence of the approximations. Finally, we perform some numerical simulations to demonstrate the accuracy of the approximations and the behavior of the discrete energy and the solution. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we develop the lower order stabilized finite element methods for the incompressible flow with the slip boundary conditions of friction type whose weak solution satisfies a variational inequality. The H1 $H^{1}$-norm for the velocity and the L2 $L^{2}$-norm for the pressure decrease with optimal convergence order. The reliable and efficient a posteriori error estimates are also derived. Finally, numerical experiments are presented to validate the theoretical results.

This paper is concerned with the convergence and superconvergence of the local discontinuous Galerkin (LDG) finite element method for nonlinear fourth-order boundary value problems of the type u (4) = f (x , u , u ′ , u ′ ′ , u ′ ′ ′) , x ∈ [a,b] with classical boundary conditions at the endpoints. Convergence properties for the solution and for all three auxiliary variables are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th-order convergence, when polynomials of degree p are used. We also prove that the derivatives of the errors between the LDG solutions and Gauss-Radau projections of the exact solutions in the L2 norm are superconvergent with order p + 1. Furthermore, a (2p + 1)th-order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages is also obtained under quasi-uniform meshes. Finally, we prove that the LDG solutions are superconvergent with an order of p + 2 toward particular projections of the exact solutions. The error analysis presented in this paper is valid for p ≥ 1. Numerical experiments indicate that our theoretical findings are optimal. [ABSTRACT FROM AUTHOR]

In this paper, we propose an optimally convergent discontinuous Galerkin (DG) method for nonlinear third-order ordinary differential equations. Convergence properties for the solution and for the two auxiliary variables that approximate the first and second derivatives of the solution are established. More specifically, we prove that the method is L 2 -stable and provides the optimal (p + 1) -th order of accuracy for smooth solutions when using piecewise p -th degree polynomials. Moreover, we prove that the derivative of the DG solution is superclose with order p + 1 toward the derivative of Gauss-Radau projection of the exact solution. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise P p polynomials with arbitrary p ≥ 1. Several numerical results are provided to confirm the convergence of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper provides an a priori error analysis of a localized orthogonal decomposition method for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in the form of the spectral gap inequality, then the expected L² error of the method can be estimated, up to logarithmic factors, by H + (ε/H)d/2, ε being the small correlation length of the random coefficient and H the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization. [ABSTRACT FROM AUTHOR]

In this paper we consider a mass- and energy–conserving Crank-Nicolson time discretization for a general class of nonlinear Schrödinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal L ∞ (H 1) -error estimates is still open, both in the semi-discrete Hilbert space setting, as well as in fully-discrete finite element settings. This paper aims at closing this gap in the literature. We also suggest a fixed point iteration to solve the arising nonlinear system of equations that makes the method easy to implement and efficient. This is illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal L 2 -norm error estimates. The approximate orders for the state, costate, and control variables are O (h 2) in the sense of L 2 -norm. Furthermore, we derive H 1 -norm error estimates for the state and costate variables. Finally, we give some conclusions and future works. [ABSTRACT FROM AUTHOR]

Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ⊂ ℝ3 with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H1(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type uxx + uyy = f(x, y, u, ux, uy), in a rectangular region Ω with classical boundary conditions on the boundary of Ω. Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1, and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal. [ABSTRACT FROM AUTHOR]

We study the problem -Δu=f, where f has a point-singularity. In particular, we are interested in f = δx0, a Dirac delta with support in x0, but singularities of the form f ~ |x − x0|−s are also considered. We prove the stability of the Galerkin projection on graded meshes in weighted spaces, with weights given by powers of the distance to x0. We also recover optimal rates of convergence for the finite element method on these graded meshes. Our approach is general and holds both in two and three dimensions. Numerical experiments are shown that verify our results, and lead to interesting observations. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a priori error estimates and superconvergence of – mixed finite element methods for elliptic boundary control problems. The state variables and co-state variables are approximated by the – (velocity-pressure) pair, and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, the state variables, and the co-state variables. Then we obtain a superconvergence result for the control variable by using a postprocessing projection operator. [ABSTRACT FROM AUTHOR]

In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified L1-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal L2 error estimates are performed and the feasibility is validated by the calculated data.