Scholarlink

<p>This paper is devoted to the numerical analysis of a system of parabolic quasi-variational inequalities with nonlinear source terms and solution-dependent obstacles. We consider a vector-valued problem defined on a bounded domain and governed by coercive elliptic operators and Lipschitz continuous nonlinearities. We establish existence, uniqueness, comparison, and monotonicity properties for the continuous system. A parabolic Bensoussan–Lions type iterative scheme is introduced and shown to converge geometrically in the strong norm <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi mathvariant="normal">∞</mi> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$L^{\infty}(\Omega\times(0,T))$</EquationSource> </InlineEquation>. We then construct a fully discrete scheme based on implicit Euler time discretization and finite element approximation in space, preserving the stability and monotonicity of the continuous problem. Under suitable assumptions, we derive an optimal <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi mathvariant="normal">∞</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$L^{\infty}$</EquationSource> </InlineEquation>-error estimate of order <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo stretchy="false">|</mo> <mo>log</mo> <mi>h</mi> <msup> <mo stretchy="false">|</mo> <mn>2</mn> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$O(h^{2}|\log h|^{2} + \Delta t)$</EquationSource> </InlineEquation>. To the best of our knowledge, this is the first optimal maximum-norm error estimate for fully discrete parabolic systems of quasi-variational inequalities with solution-dependent obstacles. Numerical experiments are provided to validate the theoretical results and confirm the efficiency and accuracy of the proposed methods.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Optimal \(L^{\infty }\)-error analysis for parabolic systems of quasi-variational inequalities with nonlinear source terms

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年5月5日

查看全文
  • Salah Boulaaras

摘要

This paper is devoted to the numerical analysis of a system of parabolic quasi-variational inequalities with nonlinear source terms and solution-dependent obstacles. We consider a vector-valued problem defined on a bounded domain and governed by coercive elliptic operators and Lipschitz continuous nonlinearities. We establish existence, uniqueness, comparison, and monotonicity properties for the continuous system. A parabolic Bensoussan–Lions type iterative scheme is introduced and shown to converge geometrically in the strong norm L ∞ ( Ω × ( 0 , T ) ) $L^{\infty}(\Omega\times(0,T))$ . We then construct a fully discrete scheme based on implicit Euler time discretization and finite element approximation in space, preserving the stability and monotonicity of the continuous problem. Under suitable assumptions, we derive an optimal L ∞ $L^{\infty}$ -error estimate of order O ( h 2 | log h | 2 + Δ t ) $O(h^{2}|\log h|^{2} + \Delta t)$ . To the best of our knowledge, this is the first optimal maximum-norm error estimate for fully discrete parabolic systems of quasi-variational inequalities with solution-dependent obstacles. Numerical experiments are provided to validate the theoretical results and confirm the efficiency and accuracy of the proposed methods.