The main objective of this thesis is to study a priori error analysis of the two-scale Composite Finite Element (CFE) method for parabolic initial-boundary value problems (IBVPs) in two-dimensional convex and non-convex polygonal domains. The main objective of this thesis is to study a prior error analysis for two-scale composite finite element (CFE) approximations of parabolic problems in both convex and non-convex polygonal domains.

## Description of the Problems

Additionally, several works involving nonlinear parabolic equations in nonconvex polygonal domains can be found in the papers by Gao et al. Therefore, in this thesis, an attempt is made to study a priori error analysis for CFE approximations in two scales of parabolic problems.

## Some Notations and Terminologies

One of the most important applications for parabolic equations with measurement data in time is related to the first order optimality conditions of parabolic optimal control problems with pointwise state constraints, see e.g. Casas [19], Lions [65], Meidner et al. The existence and uniqueness result for parabolic problems with measurement data in time can be found in the work of Gong [46].

## Framework for the CFE Method

From the definition of SCFE it is clear that SCFE ⊂ S, where the values at the slave nodes are bounded by extrapolation. i) The dimension of the SCFE is determined by the number of nodes in ϑdof. For the constant time step, in each time interval we have =k, i.e., the size of each subinterval (ti−1, ti] isk.

## Background and Motivation

The inverse Euler method is used to estimate the error in the fully discrete scheme. We show error estimates for the spatially semidiscrete problem as well as the fully discrete inverse Euler method.

## General Outline of the Thesis

The fully discrete backward Euler method (non-linearized form) has a disadvantage that at each time level it is necessary to solve a non-linear system of algebraic equations due to the presence of two non-linear terms a( Un) and f(Un). In this chapter, we derive a priori error estimates for the spatially semidiscrete and fully discrete CFE approximations of the problem (1.1) in a convex polygonal domain with smooth initial data. Optimal order error estimates (up to logarithmic terms) for the spatially semidiscrete and fully discrete CFE methods for problem (2.1) with smooth initial data are performed in Section 2.4.

## Some Geometric Constants

The key technical tool for the error estimates includes the estimates for the associated elliptic or Ritz projection in the framework of CFE method. For this, a quantity will be introduced into the error estimates that measures the overlap of such neighborhoods. The error estimates for CFEs rely on the existence of an appropriate expansion operator on the domain Ω.

Existence and Uniqueness of CFE Solution

## CFE Error Estimates

### Spatially Semidiscrete Error Estimates

In the following lemma we estimate the errors in the Ritz projection in the framework of the CFE method. Then there exists a positive constant C depends on σdist, σe, σuni, Col,1, Col,2, σext and the minimum angles in the triangle TH,h, such that. The main result for semidiscrete CFE error estimation is given in the following theorem.

### Fully Discrete Error Estimates

Now, with notations as in the semidiscrete case, we express the above equation in vector and matrix notations as ACFE+kBCFE)αn=ACFEαn−1+kfCFE(tn), where ACFE+kBCFE is positive definite and therefore invertible. We will prove the following error estimate which reduces to that of Theorem 2.4.2 for constant time steps with kn =k. In the Crank-Nicolson method, the semi-discrete equation (2.2) is discretized in a symmetrical way around the point tn−1. with U0 =uCFE0 , which can be stated in matrix and vector form as. Since ACFE+12kBCFE is positive definite and therefore in particular reversible.

## Concluding Remarks

This chapter deals with the a priori error analysis of the spatial semi-discrete and fully discrete CFE approximations of the homogeneous parabolic problem (1.2) in the convex polygonal domain, given the data u0 ∈L2(Ω). Eigenfunction expansion associated with the elliptic operator and rational approximations for the exponential are key technical tools used in error analysis of nonsmooth data. Like the standard FEM, the analogous behavior of the CFE solution is studied and the optimal order convergence (up to logarithmic terms) is derived for the solution even for non-smooth initial data.

## Introduction

Following [102], we introduce some Hilbert spaces ˙Hν(Ω), which are convenient for describing the regularity of the solutions of the given IBVP (3.1). Thom´ee [102] studied the a priori error analysis to approximate the solution of a homogeneous parabolic equation with finite elements in convex domains with non-smooth initial data and proved that the optimal order convergence takes place in positive time even if the initial data are non-smooth. In this chapter, we derive a priori error bounds L∞(L2) for semidiscrete and fully discrete CFE approximations of the homogeneous parabolic equation (3.1) in convex domains with nonsmooth initial data.

## CFE Error Estimates

### Spatially Semidiscrete Error Estimates

For problem (3.1), writing the semidiscrete problem and the continuous problem in terms of TCFE and T, we find that. To estimate reTCFEek, we integrate the error equation (3.6) from 0 toTCFEe(0) = 0 to obtain. ζ2kρtk+ζkρk+kˆρk), and this completes the rest of the proof. Now we are able to derive the semi-discrete error estimate for non-smooth initial data.

### Fully Discrete Error Estimates

To estimate the error we will look for some properties of the rational function r(λ) that approximate e−λ. We are now able to show the error estimate for unsmoothed initial data for the backward Euler CFE method. Then under the assumptions of Lemma 3.2.3 with uCFE0 =PCFEu0, where PCFE is the L2 projection on SCFE, we have.

## Concluding Remarks

This chapter is devoted to a priori error analysis for semidiscrete and fully discrete CFE spatial approximations of problem (1.1) in a nonconvex polygonal domain for smooth and unsmooth initial data. In addition, for the homogeneous equation, the error estimates for the semidiscrete and fully discrete spatial methods are set to the L∞(L2) norm for unsmooth initial data. A priori error estimation for semidiscrete and fully discrete spatial CFE methods with smooth initial data are performed in Section 4.2.

## CFE Error Estimates

### Spatially Semidiscrete Error Estimates

Since Ω is non-convex, the exact solution u /∈ H2(Ω) ∩H01(Ω), i.e. the exact solution of problem (4.1), has a lower regularity due to the re-entry of the boundary angle Γ. Let ECFE(t) = e∆CFEt be the discrete analogue of the solution operator E(t) of the homogeneous case (4.1). With h=O(H), the results of Theorem 4.2.1 coincide with the results of standard FEM for non-convex domains.

### Fully Discrete Error Estimates

This is because in this case the two-scale lattice TH,h coincides with the coarse lattice TH. We will now study the fully discrete variable time step method for the backward Euler method. The above estimates together with the standard estimates for ρn and θ0 complete the rest of the proof.

## Nonsmooth Data Error Estimates

The backward Euler scheme is shown as follows: Given U0 = uCFE0, find Un ∈ SCFE such that. We then show the error estimates for the fully discrete backward Euler method with non-smooth initial data.

## Concluding Remarks

In the previous chapters, we considered a two-scale CFEs for approximating linear parabolic IBVPs with the Dirichlet boundary conditions. With respect to time, an optimal order convergence O(k) and O(k2) has been proven for the backward Euler and the Crank-Nicolson method respectively. In this chapter, we considered a two-scale CFEs for approximating nonlinear parabolic IBVP with the Dirichlet boundary condition in a nonconvex polygonal domain.

## CFE Error Estimates

### Spatially Semidiscrete Error Estimates

Now, to estimate kρ˜tk, we again use the duality argument as in the proof of Lemma 5.2.2. Choose χ=RCFEψ in the above and use integration of parts formula for the last term we obtain. The following theorem shows an estimate of the error in the L∞(L2) norm for the semidiscrete problem (5.3).

### Fully Discrete Error Estimates

To prove the uniqueness of the solution, we first show the following error estimate which is valid for every solution of (5.16). The disadvantage of the method described above is that at each time level we had to solve a nonlinear system of algebraic equations due to the presence of two nonlinear terms a(Un) and f(Un) in (5.16). Linearized modification for backward Euler's method: Replacing the terms a(Un) and f(Un) in (5.16) with a(Un−1) and f(Un−1) respectively, we get.

## Concluding Remarks

Two-scale CFE method for parabolic problems with measurement data in time for convex and nonconvex domains. In this chapter we study the spatially semidiscrete and fully discrete CFE approximations of parabolic problem (1.4) with measurement data in time for both convex and nonconvex domains. A priori error estimates in the L2(L2) norm for the spatially semidiscrete and fully discrete problems for both convex and nonconvex domains are derived.

## Some Auxiliary Results

For the error analysis of the semidiscrete problem, we introduce the so-called elliptic orRitz projection RCFE on SCFE as the orthogonal projection with respect to the inner product (∇v,∇w), such that. For convenience, we now recall the following additional results for the errors in the Ritz projection for convex and non-convex domains, respectively (cf. Lemmas 2.4.1 and 4.2.1). Then there exists a positive constant C that depends on σdist, σe, σuni, Col,1, Col,2, σext and the minimum angles in the triangulation TH,h, such that.

## CFE Error Estimates for Convex Domains

### Spatially Semidiscrete Error Estimates

We derive a priori error estimates in the L2(L2) norm for the spatially semi-discrete and fully discrete problems in both convex and non-convex domains. Our analysis is based on the error estimates for the associated elliptic projection in the CFE framework and the duality argument. An order O(HLogg1/2(H/h) +k1/2) of convergence in the L2(L2) norm is obtained for the complete discretization of the finite element approximation of parabolic equations with measurement data in time for the convex polygonal domain, whereas the order is reduced to O(HsLoggs/2(H/h) +k1/2), 1/2≤s≤1 for the nonconvex polygonal domain with respect to the space discretization.

Estimates for the semidiscrete spatial error in the L∞(L2) and L∞(H1) rates with smooth initial data (cf. For both the convex and nonconvex domains, the error estimates for the semidiscrete spatial method (cf.